If you wired everything according to the manual, then it's doing exactly as it's supposed to. The Car EZ starts the PC, which it is supposed to do. I don't understand why you think that's a BAD thing (as indicated by your frownie face).
The problem is the PW200M.
When you're starting the car the voltage drops as it feeds the starter. The PW200M doesn't regulate the 12v line to provide constant power, especially during engine crank.
Frankly, the PW200M is a piece of ****. It doesn't provide the 200w that it says it does. If you get a regulator for the 12v line, you're going to have about 60w regulated, if I remember correctly.
I suggest looking into either a Keypower 250w unit (which has no shutdown controller) or the DSATX 220w which has all the goodies built-in.
The Opus makes a 150w PSU that includes a startup/shutdown controller.