The earliest pistol-wielding cavalry force in Europe was the German Black Knights, called "Schwarze Reiter" or shortened as "Reiter". When this type of unit was introduced to France and England, they picked up new names such as "pistolier" or "pistoleer". I suppose pistoleer is a nice generic name.
The contemporary nomenclature for cavalry units in Europe from the mid-16th century on got really, really complicated. Units like the 'Reiters' with gunpowder weapons were variously called pistoliers, or mounted arquebussiers, but by 1632 Cruso's "Military Instructions for the Cavalry" divided cavalry into Lancers (which meant the old–fashioned knights with lances, NOT light cavalry as they became later), Cuirassiers, Harquebusiers, Carbiniers and Dragoons.
Harquebusiers and carbiniers were virtually identical, since their weapons were simply 'longer than pistols' but varied from 30 to 48 inches long and their tactical purpose was identical: provide fire support fo the cuirassiers.
Cuirassiers by the early 17th century, by the way, were equipped with a sword and two pistols each, but also heavily-armored with full body armor, helmet and thigh armor, and so were very expensive - only a single regiment was raised in the English Civil War, for example.
Dragoons were mounted infantry, not trained or given horses good enough to charge with, and in continental armies of the time were frequently classified separately, as in "Horse, Foot and Dragoons" to describe an army's components.
Since everybody except dragoons could be wearing some kind of metal cuirass at some time between about 1500 and 1650 (including harquebusiers and pistoliers and carbiniers), the graphics get confusing: the pistol-toting cavalry shown in the game video could really be Cuirassiers, Pistoliers or 'Reiters' depending on the exact date or the army: The Russians were forming units of 'reitars' as late as late as the second half of the 17th century and everybody's cavalry were still issued pistols (1 or more) right into the middle of the 18th century, even though it was no longer considered tactically efficient to try to use them in battle: John Churchill, in fact, starting in 1702 only issued 3 bullets to each of his cavalrymen because their pistols were only to be used to protect their horses when they were turned loose to forage, NOT in battle!
As for the term 'Cuirassier' itself, it is also a misnomer. While it always meant an armored cavalryman, men on equally-large horses with heavy straight swords and the same tactical purpose - to charge home and break an enemy unit in battle - were also formed without body armor of any kind: the English/British 'Heavy Dragoons' were heavy cavalry of that type, without the cuirass but with every other cuirassier's characteristic in battle. The Royal French Army in the century before the revolution had only a single regiment of armored cuirassiers, the
Cuirassiers du Roi, but dozens of regiments of men on equally-large horses with equally heavy swords and the tactics of Battle Cavalry. In-game, they would have the same tactics and virtually the same factors as 'cuirassiers'.
Even as late as the Napoleonic Wars, only the Russian and French armies had large numbers of armored men on big horses as Cuirassiers. The Austrians only armored the front of their men (their 'cuirassiers' wore no metal back-plates) and the British and Prussian armies had no armored cavalry at all, regardless of what they were called.
Ironically, everybody formed regiments of armored cuirassiers after the Napoleonic Wars, just when rifled firearms made body armor virtually useless. By the end of the 19th century, except for parade units like the British Horse Guards , body armor was no longer worn by any cavalry.