AFAIK, "really believe" is used to mean both "emotionally accept" and "have as a deliberative anticipation-controller". I take it you mean the first, but given the ambiguity, we should probably not use the term. Just a suggestion.

Here's the thing: intellectual beliefs aren't always anticipation controllers. Sometimes, they're just abstract information marked as "correct" -- applause lights or teacher's passwords.

So, by "really believe", I mean, what your automatic machinery will use to make the predictions that will actually drive your actions.

This also connects with "emotionally accept" -- but isn't precisely the same thing. You can emotionally accept something without actually expecting it to happen... and it's this autonomous "expectation" machinery that I'm referring to. i.e., the same sort of thing that makes your brain "expect" that running away from the haunted house is a good idea.

These sorts of expectations and predictions are always running, driving your current behavior. However, conscious anticipation (by definition) is something you have to do on purpose, and therefore has negligible impact on your real-time behaviors.

Not sure I get the distinction you're drawing. Supposing you say you know you won't win, but then you buy a lottery ticket anyway. Is that a failure of emotional acceptance of the number representing your odds, or a failure of anticipation control?