As I understand it... it's kinda both. The teams involved do want to stay in their traditional national leagues. So the English teams would play in the Premier League. But they would NOT play in the historical Champions League.
The way (I think) it works is like this... in the past, teams played in their individual national leagues. The best X teams from individual national leagues qualify for the Champions League (there my be a qualifying pool bit, I'm not 100% sure). There are some special rules like last year's champ always qualifies, if I understand correctly. But basically, you have to do well in your national league to get into the Champions League.
The new Super League would pull all the 'best' teams out of the Champions League and lock them into the new Super League. No chance they ever fail to qualify. And a lot less spots for non-'elites' to get in. They would keep open five slots for all the rest of European soccer teams to fight for on a yearly basis.
One of the coolest parts of international soccer (to me) is the possibility of relegation. Don't perform up to your current league's standards? Drop a league. No 76ers 'Respect the Process' baloney. Even games at the bottom of the standings matter. Less chance of teams just giving up on the season. And the possibility of relegation also allows the possibility of promotion. The spirited team of underdogs you root for can (and very occasionally do) shock the world and move up a league. The golden ring is always there... just at the edge of your reach. But the proposed Super League upsets that apple cart. The top teams are locked in. Cannot get relegated. Cannot fail to qualify. And no one else can get promoted. Sure they can qualify for a stint in the Super League. But they cannot earn a spot there.
 As I understand it, some of the better leagues get four teams into the league, while some... less quality leagues only get their winner in. Americans can kind of think of the Bowl Championship Series in college football.
 Not saying everyone has to get into soccer. If you've tried watching and it doesn't float your boat, that's just fine.
 They almost always drop back down after a season or two.