No. A belief can be wrong, right?

So can an assertion. Just because you assert "snow is white" does not mean that snow is white. It means you believe that to be the case.

Technically, asserting that you believe snow to be white does not mean you do - but it's a pretty safe bet.

Likewise, something can be true, but not believed by me (e.g., my wife is having an affair, but I do not believe that to be the case).

Yes, but you didn't assert those things. If you had asserted "my wife is having an affair", we would conclude that you believe that your wife is having an affair. If you asserted "I believe my wife is having an affair", we would conclude that you would assert that "my wife is having an affair" is true.