would presume that the contract you present to the league has to specifically state what exception you're using
I believe that this is correct.
I don't think you get to choose though. At the time of the MLE contract, you are either over or not and the MLE is declared one or the other by the league. That is what the clause says, or at least my read of it. The Celtics can game the system to either be just under at the time and then you can do a S&T but you are hard capped or be just over at the time and then no hard cap but no S&T. I feel the latter is better for the Celtics. We aren't going to get Markkanen at this point anyway. Better to focus on straight trades for Gallinari or Bagley for example or some bigger fish down the road.
Well, regardless it's like the choice is made for you depending on your roster at the time of the signing. Maybe you say "I'm using the MLE" then the league sees where you are with your roster at that point and says you either used the tax or non tax one. Don't think the distinction matters at that point.
As said previously, the important part is when Schroder is signed officially... that'll tell us where we at, until then we're just using words that don't have meaning. It's more likely than not that we'll need to use the taxpayer, I think that's why the Celtics may still be looking to move some payroll (Dunn for example) to give us wiggle room.