The only multi voltage ballasts I have seen are usually multi-tapped. It has been awhile since I was out in the fields with the tools. They might have something new that I don't know about. I am heading to my supply house this afternoon and I can check in with them.
There have been, and probably still are electronic ballasts for T8 lamps that are multi voltage input. I remember 120/277 with single input (hot-neutral) factory conductors, and the internal ballast components sensed the input voltage.
I also remember (with 'new' electronic ballasts) not reading the label, and putting 277 to the 120 volt only ones! No disaster, only the ominous click, not even a puff of smoke.
The OP is wondering IF say you put 277 into a ballast, could you still use it for 120, or is it 'locked' into looking for 277?
Years ago, remember we've been using the electronic fluorescent ballasts out here for nearly 3 decades now, the multi-volt ballasts were said to have a sacrificial 'jumper' that burned out depending on the first voltage is was connected to.
That made it unusable for any other voltage after.
That was then and today it's not the case (from what I've seen and read), but there's a LOT of sparky's out there that still go by that belief. Our own warehouse has hundreds of electronic ballasts in bins labeled as good for only one voltage, how many actually are is for someone else to determine.
I also recall hearing something about this issue on multi-volt ballasts back in the early 1990’s, but I never was able to confirm it myself. I do seem to remember that some older Grasslin DT series multi-volt defrost timers I installed on walk-in freezers had what they called automatic voltage adjustment. They would work on 120/208/240. You just hooked them up and they would “burn in” at whatever supply voltage they were connected to. They must have done away with that feature, because the last ones I saw had the little DIP switches on the board that you manually set for the required voltage.