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I have noticed there is a large number of words that translate as “player” and “team”. I've been trying to decipher those and so far it seems that:

玩家 wánjiā, 玩者 wánzhě – those two mean a player of a game

竞赛者 jìngsài zhě – this one would be a contestant in a game such as
race?

游戏者 yóuxì zhě – a video game player or gamer

球员 qiúyuán – player of a game involving a ball, such as football,
basketball (golf?) etc.

And there there are the teams:

班 bān – team?

团队 tuándu – team?

组别 zǔbié – team?

球队 qiúduì – sports team involving a ball

Is my thinking correct? What is the difference between those terms? Are there other terms I am missing here?

Depend on context “player” have to be translated differently according to the adjectival description

1. team athlete: '运动员/ 员'- e.g. 篮球运动员/ 篮球员 (basketball player); 足球运动员/ 足球员 (football player)




2. individual athlete: '手' - e.g. 网球手 (tennis player); 羽毛球手 (badminton player)




3. musical instruments player : '演奏者' - e.g. 钢琴演奏者 (Piano player); 吉他演奏者 (guitar player)




4. game player : '玩者/ 玩家' e.g. 游戏玩者/ 游戏玩家 (game player);

Note:

钢琴家 = (professional, master) pianist

吉他手 = (professional, master) guitarist

• sport team is called 队 as in 上海足球队, 上海篮球队 (both can be called 上海队); 球队 just means team(s) of a sport involving a ball, as in '上海的球队' (Shanghai's teams)




• For pro soccer organization, we usually call it 足球俱乐部 or 足球球會/ 球會 (soccer club) , the 足球队 (soccer team) belong to the 球會 (club)

Your thinking is correct!
班 is always 班级 (bānjí) — usually it means "class"

团队 (tuánduì) — you might consider the word as "group"

组别 (zǔbié) — also means "group", but it doesn't always have many people in the group

球队 (qiúduì) — sports team involving a ball, it's only used in ball games, like 足球队 (football team), 篮球队 (basketball team)…