There is probably old oil from the engine that has gotten into the starter, and is causing it to gum up when it is cold.
Basically the starter spins it spins by centrifigal force the gear forward and engages the flywheel to crank the engine. If the shaft that the engaging gear is on isn't lubricated properly it won't go forward and therefore not engage.
Take it off and clean it up, lube with proper viscosity stuff and see if it doesn't solve it, or if you have taken it off just replace it, it's an easy job.
The starter drive is bad. Its the gears that engage into the engine. It like a sprag in a trans. A one way clutch. With a starter in your hands if you try to turn the teeth by hand if you turn it one way the gear will just turn when you turn it the other way the whole drive and motor will turn. What it does is it locks up when you crank the engine and when the engine starts the engine will spin faster then the starter is while you are tring to let go of the key so it allows the drive to unlock and the engine won't be able to spin the starter faster while you are tring to let go of the key.
Cold weather is a common time for them to show up bad. When they first start going bad a few hits of the key adds a little temp to it and the car will crank up. But eventually it will fail and not lock up. I haven't had a 2.2 starter apart but on most starters they are replacable. You usually have to just about completly disassemble the starter to do it.
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