They haven't been super consistent with how they refer to Champions as of late. Sometimes it's "x region Champion", other times it's "x league Champion". Shirona was called the "Sinnoh Region Pokemon League Champion" not too long ago I believe.
League winner is an entirely different term, yeah (yuushousha). Same with the term used to refer to Dande being "reigning champ" of the PWCS (ouja, the native Japanese term for champion).
Yes.
Daigo was also called "Hoenn Region Pokemon League Champion" in the second Mega Evolution Special.
Maybe "x region Champion" and "x league Champion" mean the same thing because Daigo and Shirona are both Champions of their respective region's Pokemon League, and the other name is just an abreviation. I think that's also how it works in the games, but there the E4+Champion are known as the "Pokemon League", meanwhile the anime has a separate tournament before that that's also called the "Pokemon League", which makes it confusing. Which is why the distinction between "x league Champion" and "x league x conference winner" (which is what Alan is) is important.
Satoshi, like Daigo and Shirona, has been called both, so I think it's pretty clear he's in the same position. It's just that since there was no E4 or Champion yet, winning the League automatically made him a Champion, while Kukui acted as the unofficial Champion due to being the strongest trainer in the region, just like how it happened in the games.