Use the plane's ID, like for a Cherokee it'd be PA28, Cessna 402 would be CE-402, etc. Yours would be CE-337. I think Cessnas are CE but may be just C.
None of your examples are correct. All type designators are four characters, for one, and PA28 ain't one of 'em. That's just Piper's model/series number, the type designators often differ from the manufacturer's model number. The M20R I fly has a designator of M20P, for example.
For ATC purposes:
Cherokee 140, 180, Warrior, Archer are P28A
Cherokee 235, Pathfinder, Dakota, etc are P28B
Arrow is P28R, etc....
Cessna 402 is C402
Skymaster is C336 or C337 as appropriate
The current official list is here:
http://www.faa.gov/documentLibrary/media/Order/Order_7360.1_.pdf
Assuming you are serious, that wouldn't work. The point is that they ignore my call sign, which includes the type. I don't think "Cessna 337 47C" is a good call sign.
Call them the way you are. When they ask your type, you say "Skymaster 47C is a Charlie 3 3 7."
Well, my world is the northeast, mostly
Well there's your problem.
And as I said, once they finally input you into the system, it's fine from that point on, the problem is that it sometimes takes several sectors until they do it. It's not just the type, BTW, it's destination too.
You're probably being put in by the first controller as a local flight, so the next guy doesn't have much clue what you're doing. This frequently happens between two facilities I transition on a regular basis - The squawk is the same, but it's still a local squawk and the facility I'm calling up doesn't know much about me. In this case, the controllers at the other facility are REALLY good, and they usually tell me to "Contact xxx Approach 1yy.zz and give them your type and destination" so I know they're missing it.
One trick you can use is to climb above 10,000 (unless you're near a Bravo) so that you can call Center for FF, IME they'll generally put you into their computer so that everyone can see the flight plan. Of course, out east they love to tell you to descend so they can pawn you off on a TRACON instead.