I would say that #2 is the definition I mostly use in my head. I only realized fairly recently that to a lot of people apparently being an aca-fan implies publishing about things related to fandom. I always took it to mean being a person with an academic background who brings the tools they've acquired in that world into their interactions with fandom/fannish texts. It's the ways of thinking, to me, rather than the specific acts of doing "real" research on fannish things. (I would consider myself an aca-fan according to definition #2, but not according to #1.)