Indeed, the meaning of the primed variables may be ambiguous. In general, it is just used to denote a copy of a variable which may have another value. For instance, for the variables of a current point of time, we may generated primed variables for the next point of time. But that may not always be just the next-time operator, it may also just denote another variable.

For the closure under the temporal operators, the primed variables denote the next point of time, so q' denotes next(q) and I' denotes anyone of the initial states at the next point of time. Hence, the automaton for the F-closure starts in anyone of the original initial states where q is false, and will remain there for a while, and may or may not set next(q). If q is made true, then the transitions R of the original automaton are enabled.

In the first case that you mention (with the flat formulas), the primed variables are just other variables, not related to the others.