Re: "Superman" or "The Superman" ?
If they do call him "The Superman" it'll be before they learn to trust him, then it'll be Superman as normal. I imagine people who don't trust him all the way through (the military) may just call him 'the alien', but that is straight up conjecture.
As for The Dark Knight Trilogy, they called him The Batman a lot, but just 'Batman' got in their a few times as well. Obviously, in BB with Falcone "I'm Batman", at the end with Rachael "Batman's just a symbol..."
Gordon Jr at the end of TDK yells out 'Batman!'
The kid at the end of TDKR (when The Bat blasts a building to clear the path) yells "It's Batman!"
There's more, but it's hard to remember...