When to move up in driver category.

It's not about "moving up", they simply offer different launch characteristics. The PGA Tour is almost a 50/50 split between 2 and 3. Work with your fitter and see which works best and go from there. You can certainly choose one or the other, but making sure it works before plopping down $600+ and getting the correct loft and shaft is critical.

When your fitter says it works best for your game.

There are pros that moved from the TSR-3 to the GT-2.

There isn’t a hierarchy to the models. 2 models tend to be more forgiving. 3 models tend to be easier to shape shots. 4 models help those with excess spin to lower spin a couple of hundred rpm’s.

Whichever one suits your swing at a fitting is the best one for you. I like to save the center of the face by hitting across the entire face so the -2 series works well for me. Including at fittings.

Like others have said, let your numbers decide during the fitting. I was in a TS3, then a TSi2, then a TSR3, and hit the GT2 the other day and will likely get fit into that one after a more thorough fitting. Same price either way and they both look similar when standing over the top of them. The sound exactly the same when hit so other than wanting extreme adjustability, there is no need to choose one over another if the TrackMan is pointing you towards a specific head.