I have had the same problems with my 94 Toyota Camry for the past 3 years. She has over 140k on her, newly tuned up, new battery, and heet in the tank.
Only when it is snowing here with high humidity, or windy (over 25 mph), or we have an ice storm. I turn the car over, and she makes a grunt, then I turn her over again and she starts just fine. Until this winter that is. Now she does it all the time when it is just below 30 degrees out. Sometimes she doesn't. Sometimes it takes two to three times of turning the key before she starts right up.
I have at times turn the key over but not the ignition part, run the gears through all gears, pumped my clutch, and pressed down on the brake and it helps start her. LOL
I'm am going crazy here. I was told moisture builds up in the geers of the starter causing ice to form, so that the it doesn't engage until it thaws out or breaks away.
Any suggestions will help.