Starter Not Working in COLD weather

I have an 01 Jetta w/130K Miles That did not start (or even crank)when it was -10F. Happened on a monday and I was traveling, so I had it towed into the shop. They brought it inside and juice was getting to the starter (starter problem right) Once it was inside and it warmed up, starter worked just fine. I ended up taking the car home because they wanted $750 to replace the starter, so I figured i would do it myself.

My question is...Is it possible for the cold weather to cause the starter to not engage, even if the starter is not bad? It is now 27F out and it starts just fine...or is it likely the starter will fail again very soon?

Just don't want to spend $200 replacing a starter that may not really be faulty...

I have had the same problem before. It has to do with humidity and the starter freezing from frost. Found a starter bad on a 00 2.0. replaced starter and customer came to pick up and nothing, no crank at all. pushed it inside... warmed up and started fine.