One of the things to keep in mind is the use of the term APC.
This is my take:
APC - Battle Taxi. Get you to the battlefield. Armour light to mid. Offers some protection against small arms, maybe RPGs. Not capable of engaging tanks and is typically armed with only an MG or AGL. Maybe at most, a light canon.
AIFV - APC with firing slits and maybe a bit larger canon. An early version of the later MICV, but which also had the idea that infantry should be able of fighting from under armour.
MICV - Armour middle to heavy (for a troop carrier, still light as compared to a heavy MBT!). More substantial armament - larger canon + a couple of MGs, a low vel-gun plus some MGs, a canon plus an ATGM system, or the like. Capable of engaging soft skinned vehicles, other MICVs, enemy infantry, and with the ATGMs, tanks, though not at a 1:1 ration - don't have the armour to survive that. Capable of deploying the infantry and then giving some substantial fire support.
FSV - MICV or APC variant that does not carry troops, but does carry a large low-vel gun (90mm+) to act as 'fire support' for the infantry. (Presumably in air-deployment or lighter unit situations where real tanks may not be available to fulfill this role)
Lift is just another way of saying Grav. It just describes how the unit gets to work. It doesn't say anything about the type of work that is done.
To me, we have:
Motorized - Trucks
Mechanized - APC or AIFV or MICV
Lift - Grav APC, AIFV or MICV
Now, you say the mechanized counter also has Lift on it. Oh well.... - I'd stick with the designation as the 4518th LIFT infantry and stick with the idea of Grav APCs or MICVs.
Or it could, as the other lister suggested, be a political thing....
