Ever since I bolted on my non-OEM replacement starter, I have had an intermitttent problem where the starter motor will just spin during an attempted start and not engage the flywheel.

It now happens very frequently since it has gotten colder (Philadelphia), but only when the engine is cold (in the morning or leaving from work).

When it happens, it doesn't sound like the starter motor is chewing or grinding the flywheel, so I want to believe that problem is with the starter and not a worn flywheel.
(I also believe this to be the case because on multiple start attempts I can hear the starter spin freely without engaging the flywheel and then on the 3rd try it engages. The flywheel never moves during an attempt so if I had a worn spot on the flywheel, wouldn't it still not start if the flywheel was worn and in the same position?) Someone please correct me if my premise is flawed!

Going with the premise that the flywheel isn't worn down,

Why would the starter motor engage the flywheel sometimes and not others?
Why would it matter if the engine is cold or if it is cold outside?
What does outside temperature and engine temperature have to do with whether the starter engages?
Is there a certain amount of voltage that causes the starter to engage?
Maybe it's a solenoid problem?

Has anyone else had these problems using non-OEM replacement starters?

Any advice would be appreciated...