If you don't stop your side gig, we're going to have to part ways.

If you don't stop spitting in the food, we're going to have to part ways.

Sam didn't want to leave but had to leave.

No you’re twisting meanings here. If it was just “stay if you want” that wouldn’t be firing but saying “if you want to remain here you have to stop doing X” is firing someone full stop

So let's call it at least "conditional firing". It's not "you're fired right now -- never come back" but "you're fired unless you decide to do X".