Probably. But who can keep them all straight? Especially when I’ve seen a secret Santa that was a yankee swap, and everyone was required to keep their new items on display for the year

There are so many different variations of this game. Your way actually makes more sense than some of them. I’ve seen picking names, bring a gift anyone would like and then you pick at random, a variation of that where you can either take a gift already unwrapped or one from the center, one where you bring something from home that you don’t use anymore (one man’s trash is another’s treasure), everyone brings a joke gift, the list goes on. And the more cultures are involved, the more you need to clarify the rules. And mixing them up can be part of the fun.