1 and 2 are Trotz and Boudreau for me and they're both pretty interchangeable. Trotz coming in in 2014 was exactly what that group needed and while he had a stacked roster he got them results (and obviously won a cup ...), Boudreau is maybe the most underrated coach in NHL history and while he had some pretty notable playoff shortcomings he won two presidents trophies playing completely different brands of hockey with a defense corps that was uhh ... not the best.
3 is Oates, who was below average at the tactical/team management aspect of the job but incredible at helping individual players improve their skills -- none more than Ovi. Don't think it's a stretch to say that without some of the adjustments he made to Ovi's game the Gretzky chase isn't possible. Oates also doesn't get enough credit for revolutionizing powerplays with the 1-3-1 to such incredible success that you hardly ever see anyone above the pewee (!!!) level run anything different these days.
As for Carbs vs. Lavy -- I know Lavy's last year ended poorly and a lot of people in the media (and particularly from a certain four-lettered site) didn't like the way he interacted with them. That being said, he got the absolute best for two years out of a roster that was way past its expiration date and lottery-caliber goaltending. He completely outcoached a record-breaking Panthers team in a series that was probably a Hathaway empty-netter one inch to the right away from going the other way and had the team playing very well with a 21-13-5 record through the new year in 22-23 when he lost Carlson to a brutal injury.
Carbs got us into the playoffs last year and played in a style that maximized the roster he had and I think was (slightly) better than the underlying numbers give him credit for, but ultimately the team last year got incredibly lucky/unsustainable results. There were also times, especially early, where his man-on-man system looked like complete suicide last year. I'd like to see him repeat last year's playoff make with a better process before I jump to conclusions.
He's certainly on his way though. Don't think there's been a 5 game stretch where we've looked this good since probably that December 22 run before the Carlson injury.