I've played Jak 2 only briefly, but I'm pretty sure Sly 2 is the most "platforming" platformer out of this trio, while both Jak 2 & R&C have more involved shooting mechanics and less (precise) jumping around. Ratchet & Clank does still play well and I imagine Jak 2 does as well.
Outside of gameplay mechanics...well, I still can't speak for Jak 2, but Sly 2 had a decent story and fun heroes/villains. Ratchet & Clank's story doesn't have any of the charm of the earlier games (original trilogy and future included), plus it's slapped together in a very jarring manner, because they didn't want you to see the whole movie in-game.
So I'd personally vote for Sly 2. I just played the HD trilogy myself in 2015 and enjoyed Sly 2 a lot. Maybe too much so, because I kinda played the first and second game in a row and maybe overdosed myself on them, because I stopped playing Sly 3 after the first world. Should get back to that at some point.
...and of course Sly 2 also has some great jams.
As a bonus suggestion, The Ratchet & Clank HD trilogy. The first one still kinda feels like a prototype, because it lacks some of the mechanics from the later games (strafing being the most vital one), but 2 & 3 felt fantastic to play.
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