FWIW, my modern GZ250 had a vacuum-operated petcock. Its smaller sibling, GZ125, has a "classic" petcock. I ended up changing mine for a "classic" one just because the rest of my bikes have the same, so I have already developed the habit of shutting the petcock off.
In the GZ250 petcock there is no "off" position, as it is in that position as long as the engine is not running-cranking. There is an "ON" position, there is a "RES" position -their names speak for themselves" and there is "PRI" position, that opens the petcock without the need of vacuum. That is used to prime the carb in the cases you have mentioned, when the bike has been sitting for some time, in order to fill the carb bowls and avoid that the first cranks waste battery energy because the carbs are not ready yet.