1993 Toyota Camry Starting difficulty

My 93 Toyota Camry for the past two weeks has developed a starting problem that happened overnight. This has never been a problem and to happen without any warning has me very frustrated. The problem: when I turn the key all the dash lights come on but that's all that happens. The starter does not engage nor does the engine turn over. My 1st thought was a low battery. So I charged it and tried starting. No change- same problem. Next, I replaced the starter thinking it was a bad solenoid. Still no improvement. So, I had it towed to a Toyota Dealer and all they said it was - was a bad battery. So a new battery was installed. How simple is that. After two days of very cold (below zero) weather I had the same problem. But now I discovered if I turned the key several times it would start on about my 4th attempt. Once the car is warm it starts the first time I turn the key. Now, one more added note; I checked the battery to be sure the voltage was good. It registered 12.73 volts.
When I get it running - it measures 14.9 volts - so I assume I can eliminate the alternator as a problem. The only I can think it can be is the ignition switch. Am I thinking correctly?