What is the turbo used on a 1985 Daytona 2.2 Turbo Z car?

What you have is a TB03XX. Could be a TB0335. Commonly just referred to as a T03. It is a T-1. This is a draw through or pull through turbo. As evidenced by the 3 bolt compressor cover. For a rebuild kit you need one that has the 4 piece carbon seal, and not the dynamic seal. 4 piece carbon seals will work on both draw through and later blow through turbos. The dynamic seal will not work with a draw through log manifold engine like yours. Lateral movement would indicate thrust bearing issue. Check the backing plate carefully for wear.
Good things to upgrade is the thrust bearing to a 360° and a staggered turbine seal. Don't cheap out on buying a kit. Try to avoid the $30 made "who knows where" kits.
I just bought a complete "full" kit with all the upgrades for $90 shipped from PT turbos (Performance Techniques) in San Bernardino, CA. I would be highly suspect of any complete kit under $60. They also have rebuild kits on Amazon and eBay. Avoid kits that say T3/T4 as they will have some misc. brackets and o-rings that you can't use on your turbo.
Good for you that you got it all apart without bustin a bunch of bolts off in the housing!

Thanks Todd for the help. I really appreciate it. As for the rebuilding the kit is only as good as the quality and since I know very little about the ins and outs of the turbo that info is money well spent.
we converted the engine while out to a T2 set up with the one piece intake and different distributor, electronics, 3 bar map ETC.
Engine got a 89 roller cam and lifters, a little porting and fresh internals.
It was a T-1 motor so we upgraded the head bolts to 11mm as well.
Setting boost limit to like 14-15psi.
I will give that company a ring tomorrow am. All bolts came out except on on the exhaust snail side.
Joe