I think the birthday person gets to choose, but I could imagine some reasonable limits. Like if one of the four people is actually allergic to chocolate, maybe a second-favorite flavor (not involving chocolate) could be a compromise. Or, if someone is actually making the cake, there might be some limitations based on their cooking ability or budget. But if you get into too much negotiation, it stops being a nice, fun thing, you know?
Since Jackie's limitations were so specific, I don't think she should have offered "to get Adam a cake" and then asked what he wanted, except it couldn't be chocolate or pie, etc.. I think she should have offered "to get Adam a carrot cake" or something specific like that. Then if he really hated carrot cake, he could say, "Oh, thanks, but no, that's okay." If you start out with, "I want to get you a cake! What kind do you want?" and then put a lot of restrictions on that, which are based on the personal preferences of the purchaser, it kind of sours the offer, I think. The way the OP is phrased, it sounds like Jackie was making Adam's birthday cake all about what she wanted, and that's not in the spirit of the gift, I don't think.