Every year, my partner of a decade and I have Christmas with her family. My family is Jewish and we also spend at least one night of Hanukkah with them. For all gatherings (including birthdays) the gifts are almost exclusively chosen and purchased by the one who is more directly related to the recipient. It always feels a little weird to me that their family gets me gifts, but they never receive anything exclusively from myself, it’s just very vaguely implied that the gifts are from us as a couple. To be fair, their parents give gifts as a couple (always from their mom). Also, I would never ask my partner to purchase gifts for my family in addition to theirs, they’re already stressed enough by finding something for their own family. I’ve never brought this up with my partner as I don’t want to make them feel either obligated to or guilty about implicitly declaring that they don’t want to.
It seems like you are over thinking it a bit. Just let it be more fluid, there is no right or wrong way. My wife often buys gifts for my family, the given is its from both of us. But she is out and about more often and spots things people would like, so she’s the main shopper.