All religions could believe correctly that a deity exists, but be wrong enough about the properties of the deity to merit no intervention.

We can't tell that they don't want to make themselves believed in...

The prior for a being that wants to be believed in but not definitively known seems very small in comparison to the prior for beings that want to hide as much as possible or the prior of beings that want to be known with certainty.

after all, many people do believe in them

Many people believe in some type of superbeings, but not necessarily anything close enough to what they are to qualify as "them" in their view. The real Sasquatch conspiracy perhaps needn't concern itself about believers in the fake Illuminati conspiracy.

and presumably if they really didn't want that, nobody would.

No, that doesn't follow. I referred to actual power and relevance, but in my conditional I didn't necessitate the ability or desire to mess individually with people's minds, or to affect specifically how each of them deals with the evidence presented to them.

But if we use that as an additional criterion (that they can and do want to mess individually with each person's mind) it would imply that they prefer believers to be believers, nonbelievers to be nonbelievers, and people to convert from one to another as they do actually happen to convert --- which in turn would probably mean that in this scenario there's no reason to be "religious" either, and we can safely ignore them, as our minds are pretty much their puppets already.