There are two different meanings for this trope: the one written in the trope itself, and the one that people seem to think this trope is about when listing tropes for a work.
The trope itself says: there is a story with a problem, and there is something in the fictional universe (in the case of the title, Superman in the DC universe) that may solve the problem instantly. As using that resource would cause a Deus ex Machina
ending, it is not used. It is either handwaved ("Superman is in a mission at the other side of the universe and can't help us with this") or simply silently ignored.
The problem with the trope name is that both Batman and Superman have their own specific franchises, and people seem to think that this trope means "Franchise A stays out of Franchise B". As in: Batman faces a threat that may justify a Crisis Crossover
with Superman, but it was not planned that way, and Batman deals with it alone, with no interference from Superman. It becomes a problem when we don't talk about Superman and Batman, but about characters with similar power levels, whose interference doesn't really have such story-breaking influence. "Superman Stays Out of Gotham" is one thing, but "Captain Marvel Stays Out of Metropolis" or "Green Arrow Stays Out of Gotham" is another.
- Change the name of the trope to a new name that avoids the misunderstanding by using characters without specific individual franchises. For example, "The Valar will not help against Sauron", as unlike superheroes the Lord of the Rings franchise is not divided in smaller franchises (but it's just an example, it's easy to think others in that direction, perhaps better than that).
- Create a new trope (if there is not one already) for the common usage, perhaps named as "Stay Out Of My Story!", or something like that.
- Check the links to the trope, fixing the misuses and keeping the correct usages.