I think everyone is in agreement about what "vegetarian" means apart from you.

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?