The standard position is standard chess, the other 959 are variants. So, Chess 960 includes standard chess and variants. But of course when we speak about Chess 960 we usually mean the variants.
Could we please stop calling Chess960 a variant?
The point you seem to be missing is that what you call 'standard Chess' is also just another variant. It is by no means the original game. And on a World scale it is even a quite insignificant variant. About 90% of the World's Chess players play by different rules.
What is 'standard' depends on when and where you are born.
'Chess' is any two-player board game with complete information with many different types of pieces, capturing through replacement, where you win by capturing a specialpiece designated as 'royal'. FIDE rules are one particular variant of that principle.

The point you seem to be missing is that what you call 'standard Chess' is also just another variant. It is by no means the original game. And on a World scale it is even a quite insignificant variant. About 90% of the World's Chess players play by different rules.
What is 'standard' depends on when and where you are born.
'Chess' is any two-player board game with complete information with many different types of pieces, capturing through replacement, where you win by capturing a specialpiece designated as 'royal'. FIDE rules are one particular variant of that principle.
Well, it seems to me that xiangqi or xogi are just different games from standard European chess, although their origins may be related historically. I tried to play xiangqi a couple of times when I was in China and believe me that it is a different game with little similarity. Any Chinese person who played xiangqi could easily beat me at it, just as I could easily beat a xiangqi player in European chess if they didn't play it before. Chess 960, on the other hand, is similar to standard chess and if you are strong in standard chess you will almost certainly be strong in 960 as well.

Its called Chess960 because there are 960 combinations to arrange the 2nd and respective 7th rank pieces, not that many different setups to play.
No, there are 960 legitimate setups to play with, with the King between the Rooks, with the Bishops on different colour squares etc. The number of possible arrangements of pieces on the back rank is much larger.
Of course this also has to do with the silly material the Chinese play. I could not play decent FIDE Chess with checkers chips that had the name of the chess piece written on it in latin letters. Let alone with these awful Chinese scribbles... When you play it in western representation it is not that bad (but you have to beware a Horse does not jump like a Knights).
But indeed Chess variants can be very different in character from each other. Crazyhouse is also a completely different game than FIDE Chess. In fact it is much closer to Shogi. Yet it is normally considered a Chess variant. Suicide or Losers Chess are even more different. These are not really forms of Chess at all (as there is no piece you have to checkmate). But because they are played with the same pieces that move in the same way, they are generally considered Chess variants. Compared to those, Xiangqi is a quite mundane variant!
All that is of course very different from Chess960, which is not really a variant, but a (or 960) sub-variants. That is exactly why I don't like Chess960 very much. It is very unimaginative, it still has all the drawbacks of FIDE Chess, and it lacks the nice symmetry and good initial positioning of the pieces, making it awkward to play and creating a messy impresion.
One should be careful with the term 'generaization'. It is all too easy to consider something a generalization of almost everything else, depending how general you like it to be.
Yah good point. We humans become familiar with things I am no different. Notice the qualities that people prescribe upon Chess960 and I do it all the time as well. It comes from a sense of self. No concept of generalization is required to understand the process of familiarity. Objects of thought become significant just because it is being thought, and we impute qualities upon it that serve that end. We become familiar, we generalize within that familiarity and become familiar with that generalization.
Chess, Chess960 and any other game is no different. They are subject to our sense of familiarity. Chess960 meets resistance, because it tests that sense and our prescription of qualities that we think are significant based on that sense. We reject Chess960, we love it, we hate it, we want to change it, we think up other alternatives and do anything but actually play it.
and so on it goes....
But Chess960 has no quality. Chess has no quality. We experience playing it. We enjoy the sensation.
That is all.
The process of familiarity tends to oppose curiosity. But Chess960 will not go away because it sparks our curiosity and tests our sense of familiarity in a special way. If we diverge too much from our familiarity, we reject that divergence altogether by denying it. If we stay too close to our familiarity, we become dissatisfied, bored, frustrated and deny it. Chess960 is truly remarkable not because of it's rules, but because of the experience it makes on our thought patterns, reflecting a sense of change in them. We think that we are flexible but often it is an illusion. For example, I could say to you that we humans are inflexible and too quickly reject the unfamiliar. People will then rise up and say "but no, I can learn 15x15 3D Galactic chess in 5 minutes and beat you! That person has forgotten that that their thought pattern has not moved one millimeter away from their familiarity.
Chess960 is at the boundary of this process. But ultimately it is just there for us chess players to experience and enjoy.
Cheers
Chess960 obviously is a Chess variant, just like 'standard' Chess according to FIDE rules is a Chess variant. (Actually, the original name of the latter is the Mad-Queen variant, to distinguish it from its pre-decessor, where the Queen moved the same as a King.) So the proper question is if Chess960 and Mad Queen are really just sub-variants of the same variant (like Capablanca Chess and Carrera Chess are sub-variants of 10x8 Chess).
I think Chess960 is not a single variant, as shuffling the pieces is not really part of the game (like, for instance, determining the opening setup in Superchess is, or, even more dynamically, like the gating of Hawk and Elephant in Seirawan Chess). It is really 960 dfferent sub-variants.
One should be careful with the term 'generaization'. It is all too easy to consider something a generalization of almost everything else, depending how general you like it to be. For instance, Seirawan Chess could also be considered a generalization of FIDE, and FIDE a special case of Seirawan where you have already gated and traded Hawk and Elephant.