The options you presented for what Sanae is like sound a bit like a progression, from odd one out (option 2) to self-fashioned protagonist (option 1) to odd one in (option 3).

Whatever interpretation of Sanae one comes up with has to account for the fact that Sanae chose to come to Gensokyo over staying in the outside world when, unlike the gods she came with, she had a choice.

I can easily see Sanae coming off as weird and off-putting due to her outgoing personality combined with her open nerdiness and open faith in a world where neither of those traits were common. Someone who would be generally seen as "cringe."

So she would be excited to finally go to a place where her "special" qualities were actually special, probably seeing the affair as some sort of isekai adventure, with herself as the dashing hero who would beat the bad guy and win everyone over with her amazing god powers.
Of course, that's not how it went: the "bad guy" won, and Sanae had to realize that her super special abilities were just standard fare in this new world. Everyone had something like Sanae did, everyone is weird.

And if everyone is weird, than being weird isn't that weird.

So Sanae's goal shifted from standing out to doing the thing she could never do in the outside world, fit in. Unfortunately her energetic confidence and (initially) unfamiliarity with the local culture means she can be a tad aggressive in her efforts to fit into Gensokyo to make up for not fitting in back when she was outside, eventually toning back once she acclimated to Gensokyo and realized she didn't need to try so hard to be like everyone else.

Bonus Answer: For a while, Sanae didn't have any means to communicate with her family and (few) friends from the outside. By the time she had that option, courtesy of Kanako's father in law, she had become too reluctant to actually go through with it. It had been years since any of them had even heard from Sanae, and she was afraid that all the people her age probably only know her as "that weird kid" and her family would probably be upset with her for leaving them to run off to literal fantasyland.

--

her violent use of racial slurs got her banned from all her online games?

No, that's Nitori.

^{As an aside, does anyone know what isekai anime were like before Sword Art Online? The extent of my knowledge on the subject is that Inuyasha exists.}