You went under your desk to plug a keyboard in. You are using that keyboard for work. If you don't plug it in, you can't do your job. OF COURSE this is a work related accident.

Helpdesk nurses are in the awkward position between healthcare and helpdesk, she probably has a script she has to stick to.
The "got hurt at work" box on that particular flowchart likely only has one line going you of it: "The guys from Legal will chew you out if you say anything here"

Because, even in more civilized countries, getting hurt at work lands everybody involved in a massive legal quagmire of responsibilities.

In this case you probably should have called someone in the company, usually there are designated people for all first-aid related issues.

First and foremost: I am not a lawyer.

You're in the U.S., but you don't say what state, so I'll speak from my experience (having been in your exact situation) in Ohio, but you might have a different experience (for example, California has much more pro-worker workers' comp laws).

First: report the injury to your employer. Regardless of whether it qualifies as workers' comp or not, they will need to be aware. If you have a large employer there will probably be a dedicated person on staff who will start handling these things. Additionally, your employer is most probably required to report the injury to a state-level BWC office. In Ohio, an employer is required to report the injury to BWC if it caused 7 or more days of disability (that does not just mean "time off work") or death.

Second: you'll be assigned a case-worker, talk to them. They will help you. Once you get a case-worker their job is literally only to get you help. Call them, talk to them, tell them what happened. Ohio has a huge boatload of rules about what you, your case-worker, and your employer must do.

Third: in Ohio, most injuries you receive at work are covered. Here's a human-speak version that isn't as lawyer-ey:

Back injuries – U.S. workers often complain of lower back pain caused by the strain of lifting heavy objects while at work. Whether the injury was caused by a blow, jolt, or sudden stress, or if it developed over time through repetitive movements and carrying heavy objects, you may be eligible for compensation for herniated discs, fractured vertebrae, sprains, strains and other problems.

Directly from the Ohio BWC (emphasis mine):

Under Ohio workers' compensation law, when a claim is filed with BWC, all medical conditions requested in a claim are considered alleged until a determination is made either allowing or disallowing them.

And (again, emphasis mine):

Allowances are medical conditions recognized as directly resulting from a compensable work-related injury or occupational disease. Allowances are supported by medical documentation submitted by providers. Allowances are also referred to as allowed conditions and allowed diagnoses.

If your state is anything like that, what it basically means is that the BWC (not your employer!) will decide what claims are "allowable". As an example, I tore an inguinal hernia doing a task that was loosely related to my job description, and I was in fact not obligated to do said task, but BWC took the injury as an allowable claim, and they paid for all of my medical expenses.

Edit: you say you are in California, so there's a whole (well put-together, I should add) booklet that talks about your benefits and such:

Workers' Compensation in California: A Guidebook for Injured Workers

I want to highlight this part (emphasis mine, from page 7):

“Accepting” the claim means the claims administrator agrees your injury is covered by workers’ compensation. If your claim is accepted, you will receive paid medical care for your injury. You may also eligible for payments to help make up for lost wages. To learn about these payments, see Chapter 5.

In California, much like Ohio, you don't decide if it's an eligible workers' compensation claim: the state does. So, while you may think "it's nothing", you still may be eligible for the full coverage of the medical costs you incurred (I suspect they're not particularly high, but still not 0.)

So, to answer your question:

Does merely getting hurt at work even if it has nothing to do with your job mean it’s on the employer?

While the burden to pay may or may not fall on your employer (read: employer's insurance company), neither you nor your employer make that decision: your BWC office does (read: can). Your office will weigh all factors of the case (including whether or not there is a reasonable expectation that you would have to do this type of labor on a regular or irregular basis) and make a judgment.

If by "why did this happen", you mean the response of your HMO's nurse (rather than the injury itself), the answer seems simple and rather obvious: the HMO wants your employer (or their insurer) to pay for the cost of treatment.

Does merely getting hurt at work even if it has nothing to do with your job mean it’s on the employer?

It can be a legal minefield but generally yes.

Circumstances and laws may change that to a no (your fault or company not liable) or a hell yes.

For instance:

• Did a supervisor say you may or have to plug the keyboard in?
  
  


• Does the company have an IT department that does these kinds of things?
  
  


• Are you allowed or prohibited to alter company equipment?
  
  


• Did you inform your employer right away about the incident?

Disclosure: I'm not familiar with U.S. laws regarding this.

Speaking to your question 'Why did this happen?':

Me: 30 years working as a massage therapist and Tai Chi instructor; delivered office/ergonomic safety seminars to colleges and corporations in the S.F. Bay Area.

You: primarily a desk worker, perhaps not in optimal physical condition.

The injury: Seemingly simple movements can be made inefficiently enough to cause injuries.

Unaccustomed bending/twisting of the lower back while ducking under the desk, as easy as it seems, is a recipe for strain injuries. Especially in people 30 and above who are not especially athletic.

I have to guess that you were not stretched or warmed up before trying to plug in the keyboard under the desk. It seems likely that you may not have used an efficient posture while reaching for the computer's USB port. Enough so to concentrate a load on one or more joints in your back or back/hip junction.

In my country, a government worker successfully claimed the following as a 'workplace injury':
She went on a business trip with a workmate, became intimate with him in a hotel, and was injured when the associated gyrations caused a light fitting to fall from the wall above the bed and strike her on the head.

Less tongue in cheek, but my contract defines workplace injuries as "any injuries that occur during work hours, or during the commute to-and-from work."

Most insurance companies will go ahead and process the claim and likely even pay for it, with the disclaimer that you must cooperate in all subrogation attempts. Basically they say that they're going to leave it up to both insurance companies' lawyer teams because the lawyers will know the legal ins and outs better than you, the health care professionals, or the claim handlers. Sometimes they even agree to split the cost between them. Just like everyone else said, report it asap.