Lemon party sounds innocent too but there are no lemons involved.

I thought fan service was anything added to please the fans. I do see a lot of people who associate it with just boobs/sexual stuff though.

The term covers both cases. It's just a question of what they're a "fan" of. Although I wouldn't call the trophy room in Smash a case of fan service.

More strictly, though, it's a little more towards the exploitation case than the "easter egg" case. It covers situations where something is put in for the sake of "fans", even if it detracts from the experience. It's where the creators prioritise existing fans over expanding the fandom. The Cloud Strife sword thing is an easter egg, an inside joke, not really fanservice.

I take it as something that's a nod to the camera. Then again hot chicks are fan service for certain fans as well

Fan service?

The fact that Kim Possible's work clothes don't cover her midriff. That's fan service.

Yes today this is what it means. Male and Mature Gamers are the biggest spender and mouthpieces in videogames, so it makes sense to target us.
Games grew up too like their fanbases.

Metroid the Other M was no kind of service. It was only sad. That said, I was far more dissappointed with giving Samus daddy issues and making her a weak and subordinate character than any display of tits/asses. You can have females characters that wear next to nothing because it's practical (exceddingly unlikely but possible), they don't care about what other people think (possible) or they use their sex appeal as an asset (reasonable if done correctly). However I would agree all to often people don't bother properly (or reasonably) explain the reasons why those characters wear what they do and it does become irksome.

Anyway, fan service is aimed at pleasing the fans (either in references or sex appeal) but that doesn't mean it necessarily hits the mark.

Sex appeal is a subset of fanservice but it does not span the entire space. (Too much linear algebra for me)

As long as fan service is free and not something a few developers would ask for a $$$. It should be fine.
Since most gamers are men/boys, fan service really lean more on boobs/waifu etc.

Fan service is basically anything that is done for the sole purpose of pleasing the fanbase. For example, Final Fantasy spinoffs like Theatrhytm is pretty much pure fanservice meant to excite long time fans of the series by giving them a bunch of the series best music in a rhythm game. The whole eye candy thing is just one subset of fanservice, although a very prominent one certainly.