“Vegetarian” is a word that means somewhat different things in different subcultures. If you’re feeding a vegetarian, it’s important to make sure that you know which definition of the word they mean.
In most English-speaking cultures, “vegetarian” means “someone who doesn’t eat animals.”. That includes red meat, poultry, fish, and anything else you’d have to kill an animal in order to eat.
In some subcultures, “vegetarian” can mean “someone who doesn’t eat meat”, where meat is defined more narrowly than “all animals."
For instance, in the observant Jewish community, most people don’t think of fish as meat (in part because it’s not defined as meat in the rules about keeping kosher). So, in many Jewish circles, a good percentage of people who describe themselves as vegetarians eat fish, but not other animals.
From both sides of this, it’s worth being aware that "vegetarian” is a word that’s used different ways in different communities. If you aren’t sure, it’s ok and good to ask what someone eats. Similarly, if you’re vegetarian and someone asks you whether you eat fish, it’s a legitimate question, not them being willfully ignorant about what the word means.
tl;dr “Vegetarian” is a word that’s used differently in different subcultures. If you’re a vegetarian eating with someone from a different community, it’s important to make sure that they understand what you don’t eat. If you’re feeding a vegetarian, it’s important to make sure you understand which definition of vegetarian applies to them.