What are the different basic types of trem??

olafgarten

You've pretty much covered the bases there, anything else is just going to be a variation on the ideas you have already. The only real options that are different are Kahlers (Like Floyds but they work a different way) and a Duesenberg Les Trem (basically a Bigsby for people who suffer from indecisiveness).

MartinB

The Les Trem does have a couple of advantages over a Bigsby type; lighter weight and easier restringing.  Though the latter always seems to be a can of worms around these parts! 
If I was building from scratch I'd sit it a little further back than a normal stop tail though, as it does sit a bit close behind the bridge if you like to palm mute.