If animal milk is vegetarian then it doesn't mean anything.
But in the context of a "vegetarian cat", animal milk is not vegetarian.
You could also be a (plant only) vegetarian and not a vegan if you, say, didn't eat any animal product but loved leather boots. So there is something to be said for the word meaning what it sounds like.
Ideally yes, but English doesn't work that way. I dont know about other languages, but I suspect any natural language will eventually develop these types of inconsistencies.
"I'm a vegetarian, pass the salmon."
Vegetarians don't eat the flesh of animals, and it's always been like that. Eggs and milk are animal products, not the flesh of animals - so vegetarians can still eat them. Vegans don't eat animal products at all, which rules out eggs and milk. Pescatarians eat fish, but they aren't vegetarians.
So if N is the number of distinct foods you can eat, N(pescatarians) > N(vegetarians) > N(vegans).
Is there anything in particular about this that you find difficult to understand? I can model it as an ontology for you, perhaps?
I'm pretty okay going with the assumption I started with though, that if someone says they are vegetarian it doesn't mean anything. Maybe it means they eat chicken if that's all there is. Maybe not. Maybe they eat dairy. Maybe not.
It's not so simple.
"60% Of Vegans Would Try Lab-Grown Meat, Says Survey"
https://www.plantbasednews.org/post/survey-says-over-half-of...
It seems that vegetarians (or even vegans) object to the way meat is produced. Not to the idea of eating flesh as such.
Don't get me wrong, I think vegans and vegetarians can sometimes be annoyingly hypocritical and arbitrary, I think I see what you're trying to get at, but as for what constitutes a typical vegetarian diet I don't see why it's so hard to see the difference between milk and meat.
It's just way easier for me to say I'm vegetarian "but I eat fish sometimes". People get that.
And speaking personally, vegetarianism is morally inconsistent, too. Just leave the whole animal alone!