Does a change in part number necessarily mean a change in th
YES, absolutely. Whirlpool doesn't change parts and part numbers unless there is a material difference with them somehow. Very often the new part will sub-over and replace the old part on a one for one basis, but if the part is changed, it usually gets a new part number.
I just consulted my Whirlpool parts catalog (from 1992). The K machine is a 1982 model, and the M a 1984. Since yours is a K2, there is little or no difference with the major parts that are listed in the catalog reference (not every part is listed). The two machines use the same timers, motors, pumps, etc. If you had a K1 or a K0, a different pump was used from the K2 or the M0 (thus a different lint filter too) but you have two the same.
Just for reference, nearly all belt-drive machines use interchangeable transmissions. They were revised over the years for sure, but other than the rare late 60s to 1973 super tall post 18lb. machines, the transmissions interchange. Even more-so for the motors. A current 285222 motor, which is essentially what you have in both your machines, will fit ANY belt drive, 1, 2, or 3 speed. In a 3 speed the extra-slow doesn't work, but at least there is something out there for what would otherwise be an SOL 3-speed machine. I'm digressing...
Brian - Your machine has a lot of common parts. The timer, motor, and pump can still be had new. Keep the 'parts' machine though - you may need a tub (NLA), gearcase, etc. though your transmission sounds great based on your videos.
Gordon