You’re correct that vegetarians don’t eat fish. People who eat fish but no other meats are called “pescatarians”, as someone mentioned earlier.

However, “vegetarian” refers to people with diets of mostly plants and sometimes animal byproducts that don’t require killing the animal (like milk and eggs).

https://en.wikipedia.org/wiki/Vegetarianism

The specific subset of vegetarians who don’t eat or use any animal byproducts are called “vegans”.

https://en.wikipedia.org/wiki/Veganism

It can be confusing because there’s sort of a tree of consumption/diets and there are a lot of terms, and some of the terms sound similar. Hope this helps.