I don't think the problem is that there are people using the term "vegetarian" loosely.
I think the problem is that there are people who forget that chicken broth is not vegetarian.
I don't get how they can do it, but they do.
Sure, there are people who a fish-ok vegetarians, but that's not the REAL problem.
You can say, "I'm a vegetarian, I don't even eat fish," but it's sort of annoying that you have to say, "I'm a vegetarian, and even chicken broth or beef broth is a bad idea."
As for what to do in the moment, I think you just say, cheerfully, "Oh, that's too bad--chicken broth is one of the things I can't eat." And then you don't eat that food but fill up on all the other ones you can. And if they're discomfitted because you're not eating the food they prepared for you, then maybe they'll learn, and mention it to someone else, and some OTHER vegetarian will be spared the steal chicken broth.
(and every time I typed "broth," I ended up with "brother.")