Ah, so you've never actually disassembled the starter and solenoid for cleaning. What's happening is that the factory grease - and whatever you used - are mixing with the dirt. A lot of them arrive on these shores already dirty. Working the tractors just adds to it. Regardless of the source, dirt reduces the lubricating properties, and then cold temps keep the goop too stiff to slide properly. Take starter/solenoid off the tractor and to an auto electric shop. If you've never stripped one of these down before, the auto-electric shop will be money well spent.
Your use of the word "strong" suggests you're not interpreting correctly the low voltage issue I described. This can only be confirmed or denied with a voltmeter. I'm not talking problem between battery and starter, rather a known problem between the keyswitch and the solenoid. I suspect low voltage on the light gauge wire that attaches to the small post on the Chinese starter solenoid.
//greg//