Both approaches are correct. Counting letter names (staff positions) tells you whether you're dealing with a 2nd, 3rd, 4th or whatever. Then you can complete the name of the interval either by counting semitones or by relating it to the major scale starting on the lower note. (Read that carefully. The major scale starting on the lower note. Not the major scale of the key the piece happens to be in.)

C to E is a major 3rd. E is 4 semitones up from C, and E is the 3rd note of C major. Two routes to the same answer.

Note spelling matters. Interval naming is about the grammar of music notation, not just the sound. Just like English, where 'but' and 'butt' SOUND the same but have different spellings and meanings. C to F# (6 semitones) sounds the same as C to Gb. But one's an Augmented 4th, the other's a Diminished 5th. (And this ambiguity opens the door to a whole fascinating area of 'theory'. Enjoy!)

(Don't try to extend the second method into minor scales. It doesn't work. The second note of C minor is D, which is a Major 2nd above C. Plus other complications. Just don't!)

Simplistically speaking, there are two common 'sizes' of third - a major third, which is four semitones, and a minor third, which is three semitones. So when someone says a 'third', they could mean either a major or minor third. (There's also a diminished and augmented third, but they're encountered less often.)

For a third, count three staff positions.... That's it. No key here.

Aah - but the staff does have a key signature. So to see how big your third is, you have to look at the key signature of your staff, and see if the lower or higher note of your third are sharpened or flattened according to the key signature (and, of course, any accidentals the notes have).

This can be confusing. I'll try explaining it my way just to see if it helps, not to say that the existing answers don't already cover the answer, because they do. But this stuff can be hard to grasp.

The size of an interval is constant in all keys.

A major third is always four semitones. A major second is always two semitones. A minor second is always one semitone. A perfect fifth is always seven semitones.

The name of an interval depends on the names of the notes used to describe it

If you're trying to identify an interval, the number of semitones will always give you more than one possibility. For example, a three-semitone interval could be a minor third, but it might also be an augmented second.

You can tell by looking at the names - if the lower note of the interval is written as G, and the upper note is Bb, you call it a minor third, because G A B is three notes, three degrees of a scale (which scale doesn't matter, we know that most scales have a different letter for each degree).

If the lower note is G but the upper note is A#, it's the same pitch (in most common tuning systems) but it's now describing an interval of a minor second.

Intervals describe keys, keys do not describe intervals

In the key of G major, you know the root position first chord triad is G B D. Why? Because the root position major triad is formed out of three intervals with G as the root of them all: a unison, a major third and a perfect fifth.

In C major, the root position first chord triad is C E G, for exactly the same reason, you just build the intervals starting on C instead of G.

In Bb major, it's Bb D F, again for the same reason, but this time you're starting on Bb.

In this way, you can say what notes are in a key, what a key is, based on the intervals from the root note. Intervals tell you how to build chords, how to build harmony in a given key, using rules which really don't care what that key actually is. Sure it's different for major or minor keys - but that's because you use different intervals. D minor is still built the same as E minor or B minor or F# minor, in terms of intervals. Depending on your instrument they may sound quite different (different resonances, timbre, even the intervals coming out different due to unequal tuning), but on a theoretical basis they're all made the same.

It's easy to start interval examples on the tonic, but it's not required

Thus: a major third is a major third whether it starts on the tonic of your current key or not.

C to E is a major third in C major, but also in D major and in A major and in F major and in F# minor. Now, you may not be very likely to see a C natural in D major seeing as how there's a C# in the key signature, but if you did C->E would be a major third, even though it starts on the key's flattened leading note.

If your head hurts a bit, that's normal

My brain melted when I first learned this. It's perfectly normal, and one of the reasons why people doing ABRSM music exams dread hitting the point where they're required to sit their Grade 5 Theory to progress in the instrumental exams.

Intervals are confusing because, like many things in music, we tend to use quantitative and qualitative measurements interchangeably.

Counting semitones is a quantitative measurement, as it's something we can measure directly. As you're probably aware, sound is vibrating air and pitch is the frequency at which the air vibrates. A distance in semitones is the ratio between the frequencies of these vibrations, where every 12 semitones (an octave) means a doubling in frequency. No matter what key or scale you're playing in (or out) this measurement will always be the same.

Interval names, with a number and quality, are a qualitative measurement. These names correspond to a specific number of semitones, but more than that they provide a description of how they sound in the relation to the rest of the work. They describe some part of the function and feel of an interval in addition to the pure physical ratio in frequencies.

As you've already noted, the numbers in the interval name (1st, 2nd, 3rd, etc.) match the count of notes in a (diatonic) scale. So, in C-major, the interval between F and G is called a 2nd, as G is the second note in the scale (F# is not in the scale, so we don't count it for this). But there's a difference between the intervals F->G and E->F. Although they're both moving to the next note, E->F is one semitone while F->G is two.

So, to distinguish between the two intervals we call the smaller one the minor 2nd and the bigger one major 2nd. This holds for most intervals: major is one semitone larger than minor. The exceptions are the 1st (unison), 8th (octave), 4th and 5th, which the Ancients deemed to be so consonant they called them perfect.

Ok, so we can name most of the intervals in a key, but sometimes we want to do something a bit different, raising or lowering a note just a semitone, even though the resulting note might not actually be in the key. We can already go one step lower from a major interval to make a minor interval, and one step higher from a minor interval to make a major interval. But what if we want to get one step higher from a major or perfect? Or one step lower from a minor or perfect? We call these augmented and diminished intervals, respectively.

"But wait," I hear you ask, "isn't an augmented 2nd exactly the same as a minor 3rd?" They have the same pitch, sure, but this name doesn't really describe the function of what we did. We raised (augmented) a major 2nd interval to get to this note, we didn't lower a major 3rd. And this is how it will feel to the listener.

Note that this is very similar to the difference between a C# and a Db: a C# is a note where we could have otherwise used a C, but we used a note one semitone higher instead. A Db is a note where we could have otherwise used a D, but we used a not one semitone lower instead.

In conclusion: semitones describe an exact relationship between two frequencies regardless of context or key, but say nothing about how a note is used. Interval names will tell you a bit about how the note relates to what you would have expected and how it will feel to the listener. This is of course rather vague and a lot will depend on context and interpretation. Does this 4-semitone interval sound like we're just playing a major 3rd, or does it feel like we lowered a perfect 4th?

A few more notes: all of this semitone stuff only really applies to twelve-tone equal temperament tuning, the standard tuning system of all modern instruments. Different tuning systems (Pythagorean tuning, for example) don't map semitones, whole tones and intervals in quite the same way.

In regards to quote from your book: "to find the perfect 5th of a note, you can count up five notes from another note staying in key", this is wrong. You'll always find a 5th, but not always a perfect 5th. For example: if we're in C-major and start counting notes from B we end up at F, which is only 6 semitones from B (not 7), making it a diminished 5th (not perfect).

Confused about intervals and keys, does the former depends on the
latter?

It might be better not say 'depends on,' but rather think in terms of levels from generic to specific.

Distance

A the most basic level you can measure an interval in semitones or as a ratio: ex. 7 semitones or 3:2. This is just a distance with no other real musical identity.

Class

At the next level we can view intervals as a class within a set of members: in the case of western music the set of letters A B C D E F G where A to E or B to F are a the interval class fifth. The is where we can define scales as steps and chords as built from thirds.

Quality

Finally, at the level of a key we have tonics and chord roots from which we measure interval classes and specific interval qualities. This is the level where enharmonic spellings become important, because they only make sense in relation to keys and chords built in thirds. This is where we can be specific and enharmonic spellings make identities in a key clear. Things like: A to E is a diminished fifth in B flat major (the key signature is where the accidental is written,) or A flat to B natural is an augmented second rather than a minor third, and other picky spelling issue like that.

For a major third, count four semitones. That's it. No key here.

That's right. You really don't need to know the key. We get the interval class from the letters and then the quality of major third by the number of semitones. That will always be true regardless of the key.

"to find the perfect 5th of a note, you can count up five notes from
another note staying in key. The key is important here because if you
count up five notes from any other note you should land on a 5th of
the original note but not necessarily a perfect 5th"

If that quotation is correct, I think it's plain confusing.

Simply put, to find a perfect fifth above any note, count up 7 semitones.

But, whether or not the note a perfect fifth above is in or out of the key, is another matter. From the quote it isn't clear if the point it to find a perfect fifth, or go to the fifth above in the key.

If you are in a key, you cannot simply count steps. You really need to know which degree members of the key are involved. There are several naming conventions. For example two semitones below a tonic is a tone that can be labelled: b^7, supertonic, TE in Solefege, or even a m7 if it were above the tonic. In a major key the supertonic is a diminished fifth above the mediant. In a minor key the supertonic is perfect fifth above the mediant. Etc. Etc.

If you spend time reading, writing, and analyzing notated music in keys, you will internalize all the little facts of intervals relative to keys. But that knowledge won't be simply interval names. It will involve understanding the various degrees within the keys.

A lot of good info written on this question. One thing I do not see is any mention of the interval pattern for the different scales. For example, the template for all major scales is 2212221. The template for a standard minor scale is 2122122. Each scale type has a pattern of semitones. Take the major scale of c. C D E F G A B C. Chords are stacked thirds. C major is C E G. So here at the conceptual level there is no mention of numbers of semitones or chord quality (major minor etc). Just count in 3’s. C is CEG, D is DFA, E is EGB, G is GBD, A is ACE and B is BDF. See the pattern? Now you overlay the semitones template (in this case 2212221) So for a c chord, C to E is 4 semitones. E to G is 3 semitones. This the standard pattern for a major chord, a major 3rd and a minor 3rd. The pattern for a minor chord is a minor 3rd and a major 3rd, a diminished chord is a minor 3rd and a minor 3rd, and an augmented chord is 2 major thirds. This is just the basics. Music has a lot of counting but it is not advanced math. Just counting. Good luck.