Just as you can define multiple barbarian tribes, each with a unique ability, as variants of a single Barbarian class, you could define multiple elite orders as variants of a single Cavalier class. The unifying concept is a bit of arrogance, a sense of being at the top or at least high up in a natural order, and a horror of being seen as unworthy of that honor (as well as anger towards those who insult their honor.) All such elite orders never check morale or save vs. fear from mundane threats, but do check morale or save vs. fear when confronted by their arbitrary "shame triggers". Failing the roll causes elite NPCs to retreat or flee and elite PCs to be shaken (Move 3, always go last) until the end of the combat or situation. Success causes elite NPCs to be enraged ("How dare you insult me thus? Have at you!")
As mentioned already, cavaliers specifically are elite horsemen (mounted knights.) Their unique ability is related to their horsemanship: never fall from a horse by accident, mounts trained for at least a year have high intelligence. There can be equivalent aerial knights (tarnsmen) or aquatic knights, or kilted yaksmen or elephant riders or other mounted groups. Other elites, with abilities linked to something other than a mount, include:
Great Sword Knights: Specialize in the claymore or some other form of great sword, instead of the lance; they may ride to battle, but don't specialize in mounted combat and usually dismount before wading into the thick of things. In combat, they act as if under a Haste spell (which I interpret as "strike first and at +2 on the attack roll") and trigger morale checks in normal men because of their ferocity. This makes them similar to the barbarian berserkers, although outside of combat, they would have courtly manners and an established place in civilized society. This variant can also be used as a base for other melee specialists, such as a unit trained in fighting with two swords; any of these elite groups can use any melee weapon, but they have an official preferred weapon, and may feel dishonored if they are disarmed while using that weapon, or otherwise lose their favorite weapon in combat.
Dragoons or Musketeers: Like the Great Sword Knights, dragoons ride to battle, but specialize in gunpowder weapons, which they use dismounted. They will thus only exist if gunpowder exists in your game world. Dragoons don't suffer accidental gunpowder mishaps with pistols or arquebus/musket, although grenades and petards are a different matter, and curses, magic, or other deliberate attempts by an opponent to trigger an explosion have no special exemption.
Myrmidons or Pikemen: Definitely not trained in mounted combat, but in infantry combat and fighting in formation. They get a +1 to their attacks and defense when fighting in a group, over and above any other formation combat bonuses involved. They also get the +1 when saving against magical or unnatural fear, in addition to the usual fearlessness of mundane threats.
Marines: Essentially, the civilized, upper-class equivalent of barbarian sea raiders. Know how to swim, and have both increased lung capacity and a better chance to avoid drowning (+30%, +6 on d20, or +2 on d6 checks for either.) They still must remove any metal armor worn quickly, but at least they don't immediately die if they fall overboard while wearing plate armor. Won't accidentally fall overboard or fall from rigging; any fall must be deliberately forced. Also know how to sail and row, but note that this distinguishes them from the marines mentioned in the LBBs, which perhaps are just 0-level mercenaries with a marine specialty. To distinguish the two, perhaps the elite version should be called "Royal Marines" or something similar.
There are, of course, many other variants possible in a given setting, based on Marines with alternative types of transportation (Barsoom-style flyers, for example) or Dragoons/Myrmidons with a different weapon or tactics focus.