Vegan-ness is always defined individually, since every vegan decides where to draw the line or else will not be able to eat or use anything (even fallen fruit has small insect life on it, for example). I've known vegans who didn't get a driver's license because the photo used gelatin in the process, and I know vegans who will buy used leather shoes. Everyone does it their own way, there is no "vegan court" that decides what is okay or not.
I agree that the AI is not able to tease apart vegan and vegetarian. Keep in mind AI answers are also always trying to "make you happy" and in doing it never poses your statements as plain wrong.
I live in coffee country, grow coffee, make coffee, unless they are adding some sort of flavoring (condensed milk? i'm really stretching here to think of anything) to the roasted coffee I can't see how any animal product would be involved. One exception might be bone/blood meal fertilizer, but as mentioned above, that isn't mentioned for any agricultural product I've seen. And if this were the case, the term would probably be "veganic" rather than vegetarian.