A few thoughts on this:
I totally understand what the terms "vegetarian" and "vegan" mean, and would do exactly what the OP did if someone said they were either of those - i.e. give them something that would be suitable for the strictest vegetarian or vegan. However, I wonder if what we really need is more commonly agreed terms for those various shades of almost-vegetarian and almost-vegan, because I've certainly known an awful lot of different varieties.
I've known someone who describes herself as a vegetarian but cheerfully admits she's not a very strict vegetarian, in the sense that she doesn't check labels for stock, gelatine etc, she just doesn't want to see meat or fish on her plate. I think she's "vegetarian" for taste and texture reasons, not ethical reasons. I also knew a vegetarian who brought up her children as vegetarian at home but didn't mind them eating fish at other people's houses (although she didn't eat it herself) simply because she knew some of their friends' parents just wouldn't be able to get their heads round cooking for total vegetarians. Both children grew up to be non-vegetarians by their own choice.
Another person said he didn't like meat, but cheerfully ate sausages and ham - he just didn't like a big slab of meat in its simplest form. He was from another country and I've since found out that it's quite common in his country to mentally separate "meat" from "meat products".
I know about 6 vegans, all of whom quite genuinely avoid eating anything vegans shouldn't eat, except that one of them says she wouldn't completely rule out eating honey, which vegans technically shouldn't have. She's quite apologetic about this, though, and would never ask for it after having described herself as a vegan. At least two of them were vegetarian for ethical reasons but became vegan for health reasons because dairy products and eggs aggravated conditions they had. Only one of them is ever in the least preachy about being a vegan, although the one I know best is very happy to talk about it if you initiate the conversation - in fact I ask him lots of questions because I'd like to be able to cook nice dinners for vegans!
I would want all of those people to be able to get what they wanted but I guess I would expect them to express themselves unambiguously when saying what they could have, otherwise they can't really complain when people get it wrong.
Personally, I think I could live as either a vegetarian as a vegan, although I have no strong urge to be either. But if I gave up both meat/fish and dairy products I think I'd miss the dairy products less.