Due to how there is a nebulous and undefined God Cycle that reincarnates Asgardian gods (and maybe even their histories), are Thor and the Asgardians fundamentally the same beings as when they appeared originally?

Or is Thor a variant of the same Thor, similar but different, from when he first appeared? If so, how many versions of Thor and the Asgardians have there been?

I would say that until Oeming's Ragnarok, the Thor from JIM #83 to then was probably the same god, as were the Asgardians, even though there were touches of Ragnarok prior to that which were averted.

After being reborn, Thor has died (Fear Itself) and been reincarnated again. By my count, that would be three different Thors, if we exclude that Beyonders/Doom stuff (which I frankly didn't pay much attention to outside of the Thor / Odin bits). And if a death outside of Ragnarok has the same impact on their rebirth.

This is a way or reconciling divergent writing, and may not be intentional, but Fraction suggested that there are indeed many different versions of the gods. For example, when Amora was splitting open the World Tree's roots, the gods (even Hela) were remembering history different -- specifically as it relates to Skurge. At least one of the gods remembered Skurge as a bard; another remembered him a shoe cobbler (IIRC).

According to this, the gods can be reborn and be changed, and their alternate histories will fuse seamlessly with the current timeline. As Karnilla did with Ulik/Tanarus when Thor was deceased.