Wow, this is why having people who don't seem to communicate well is no fun whatsoever.
I can't tell from what has been posted if your FMIL has been stingy on the details because
1) That's just how she is and it doesn't occur to her to tell people things / she assumes they know what they need to know.
2) She specifically doesn't want her Ex to know about Steve (for whatever reason).
3) She plans on telling but hasn't got around to it/is waiting for a chosen time etc.
So here is my 2 cents about how to navigate between people who have different levels on information, when you aren't sure why they aren't on the same page: Stay as neutral as possible! You don't want to provide anything that isn't specifically asked for, nor do you want to hide anything. Maybe this sounds too complicated or like you are half-lying, but IMHO it is the only way to stand in the middle when you are put on the spot and don't know all the facts.
So if FFIL says FMIL is moving to whatever place for X reason, and you understand her to be moving because of Steve, you don't say so - you say something like "Well, I don't know all the details but maybe she has another reason to go there. If you are concerned you better talk to her about it." You don't want to encourage misinformation, but you also know it isn't your place to enlighten your Dad (and you might not know everything anyway).
If your FMIL is lying, you should not help her. However as OPs have said you don't have to go out of your way to tell her Ex about Steve. Just avoid the gossip loop whenever possible, and direct FFIL to talk to FMIL about his concerns so she can choose to tell him about Steve - or not.
Hypothetically, if your FFIL asked you point blank about your FMILs relationship
status, I would again tell him "You'll have to take that up with her."
Similarly, if FFIL says anything that takes your mind into (potential) Steve territory, just say "You'll have to talk to Mom."EDIT since reading update post:
Okay since this is affecting you, you need to take your concerns to your FMIL and ask her why she is telling your FFIL one thing and yourselves another. Consider telling her that this is making it difficult to talk to FFIL and that you are not comfortable supporting her misinformation. Ask her if she can either find a way to break it to him or leave you out of it. Just keep encouraging them to talk to each other instead of you!