Lets hope that it is the starter and nothing else. I replaced my starter a week and a half ago and it had that funny whizzing/squeeky noise when the engine fired up and would stop as soon as the starter disengaged. I posted a topic about this at the same time. The car fired up all the time without a issue and ran great except for the ignition warning light which would come on periodically. I was about to see if I could tighten the belt up a little today to see if that was the issue with it, otherwise I would put a new alternator in.
Anyway I parket my car yesterday afternoon and decided to put it in the garage a few minutes later. The car started up with the usual whizzing/squeeky noise, and stop as soon as the starter was disengaged. A few seconds later there was this loud squeeky noise which sounded like a fan belt or something, so much so that my wife came out and asked what the noise was. I rev'ed the car up and the noise stopped. I turned the car off and then on again and all I heard was the click click when I turned the ignition key to start up.
This is the second starter I have put in since summer, both used, one from a parts car and the other and more recent I bought at Stowe which bench tested ok. Battery is fully charge.
About the throw the towel in :-(