OK...let's see if I can't solve this once and for all...
A Main Battle Tank is designed to be able to suport infantry, engage other tanks, and assault fixed positions, all with some measure of success. Main Battle Tanks require a good balance of speed, armor, and firepower. MBT's can take plenty of punishment and keep fighting.
A Main Battle Tank carries at least one seperately-mounted MG (not a coax) for defense against infantry and an even mix of HE, HEAT, and capped AT rounds to deal with anything it may face on the battlefield. Almost all MBT's have a rotating turret.
An Assault gun is designed to do essentially the same things as a main battle tank, but since they are not so heavily armored and are usually quicker, they can excel as tank destroyers, at least when they have a powerful cannon. Most assault guns have an MG mounted in the front of the hull. They rarely have a closed turret, usually being mounted in a casemate, with a limited traverse. Once again they carry a mix of shells to deal with multiple threat types.
A true Tank Destroyer is what we're after in this GB. Tank Destroyers are vehicles that are either designed for tank killing or are highly successful at it. A TD first and foremost must have a very powerful weapon, and secondly must be very quick and mobile. Armor protection is usually an afterthought, since they are intended to support heavier vehicles. The only very heavily-armored TD's are the Jagdpanther and Jagdtiger, and while they were powerful, both (especially the Jagdtiger) were logistical nightmares. Tank destroyers carry a dedicated anti-tank weapon that usually has poor HE capabilities, and 90% or more AT shells. Tank destroyers rarely have MG's and when they do they usually have less than 300 rounds of ammo because of space limitations. They usually employ fire-and-move or ambush tactics when fighting on their own, unless in great force (i.e. the Russians)
Given the above, the Firefly is a tank destroyer because it excels only in that role. It has poor anti-infantry capabilites and no MG.
The M50/M51 Super Sherman, on the other hand, is a Main Battle Tank becuase it excels in all roles. Tank Destroyers were invented to support medium tanks before the concept of the do-everything MBT, which is why there are very few TD's seen after WWII.