I suppose some people do do the "lift the bellows closed" technique, but I've always felt that to be a bit too showy, and it's never really worked for me besides.
I think of the bellows as having an imaginary hinge on the bottom (in fact, you can temporarily
leave the bottom strap snapped to get a feel for what I'm talking about). I open the top of the bellows and let it sort of fall open (aided by my left hand) in a V shape. I close it by imagining that I'm pushing the V back shut.
In practice, I don't really get a "perfect hinge" due to the mechanics of the bellows, but the point is that I don't want to wind up looking like this guy
, pulling the thing straight out and snaking it back.
Checking out some YouTube videos of good players will help. Here's one of Phil Cunningham
, for example. Or, since you're jazzer, here's Frank Marocco
. In both cases, notice the V on the open, and how they don't really lift the LH side up above the RH side when closing... it's more of a push than a lift. You shouldn't need the hand strap to be tight to pull it off (in fact, you should barely need the hand strap at all when closing.)