I don't know how to go about what I have to say, so I'll just start at the top.
The CDM9 and it's variants are swingarm mechs (they work on an arc, powered by electro magnetic coils), while the CDM-12 series are linear (they rely on a cog and motor system, and move in a straught line).
I only know of around 50 players that used the CDM9, this being mainly due to cost limitations on budget players, whereas CDM-12's are fitted in well over twice that number.
Both are used in high end machines, though you would rarely find a CDM9 in a budget model.
The CDM-12 is available in a number of models, ranging from the budget 12.1 etc. up to the CD PRO and PRO2 (also M and LF versions).
I could go on, but have a look round the net, you'll get way more info than I could write here.
One last thing, stick with the Philips based machines, there are so many machines based their lasers and dacs etc. that mods are eaily found and spares are plentiful (and for the most part, pretty cheap), but watch out for people selling parts as "brand new", especially for lasers that have been out of production for many, many years.
The CD960 uses a CDM-1 and the CD880 uses a CDM-1 mkII. Both are swingarm types, but far superior in build, especially the CDM-1. (see my Ebay auction for one I have for sale. mscwilson Ebay.co.uk)
Ive just bought a CD960 and they are one of the best machines they made. Keep hold of yours.
The CD850 mkII uses a CDM-4, also a swingarm type. Can be upgraded to a composite mech (unless you already have one fitted, and even to a cast alloy version, the CDM-4/36 pro metallic (used by Krell. Same mech and laser).
Finally, don't pay $$$$'s for yopur repairs, just cos the player cost a fortune.
Nine times out of ten you'll have spent hundreds or thousands on a player, only to have the same laser fitted as in one costing a few dollars/pounds, e.g GRYPHON TABU CDP1, mega money player, fitted with a £10.00/ $15.00 KSS213A laser.
CDM-12.4 are available on the internet for under £15.00/ $25.00.
Don't get me wrong, the work that goes into most of these players is stunning, and what they get from these lasers is nothing short of breathtaking, but just be sure you don't pay over the top for your repairs, that's all.
Best of luck to you all.