Essentially what you've said is correct, except that terms like "First to act", "First position" and "Under the gun" can still be used when playing short-handed.
At a "full ring" (9-handed) table, an alternative term for the player before the Cut-off is the "Hijack". Likewise, "Lojack" is sometimes for the previous player (first to act). So in "6-max" you have the following:
- Under the gun / First position / Lojack
- Second position / Hijack
- Button (or Dealer)
- Small Blind
- Big Blind
With five/four players at the table, the "Under the gun" player is effectively also the Hijack/Cut-off; "Under the gun" is perhaps more commonly used.
With three players, it's most common to use the terms Button, Small Blind and Big Blind only.
In heads-up poker (i.e. just two players), the Dealer becomes the Small Blind. In this scenario it's most standard to use the terms Button and Big Blind.
Which terms you should use for 4 to 6 players depends on context. When discussing a live game in play, where everyone can see the action, terms like "Under the gun" are fine. But when taking notes or writing about a historical hand, this can be confusing unless it's clear exactly many players are playing.
So when taking notes, describing a hand or just to avoid ambiguity, I suggest using the full ring terms - so in 6-max the first player to act is the Lojack, and at a 5-handed table it's the Hijack. But if you use "Under the gun" then someone watching the game will know what you mean.
I don't know of any "different" names for the positions, but there might be some slang terms used in some poker communities.
Caveat: all of the above may vary regionally.