Please don't have that pop up every time you start. After a game is installed and working properly, when I double-click a game, I want it to start immediately, and not have to fiddle with other options.
In your sample dialog, this could be fixed by having a checkbox for "Always play this version". Each game version should also have an in-game option to allow the player to get back to the choice dialog the next time they start it.
At least for Steam, it is also possible to have the context menu have multiple choices for a single game. For example, here is a screenshot (with colors modified to make it easier to read) of the right-click menu for the Steam game Commander Keen Complete Pack (which actually isn't all the Commander Keen games (grrrrrr), but now I'm getting off-topic).