Here are the Historical Options:
"Heavy Cavalry" or "Battle Cavalry" as it was starting to be called in the 18th century CE, always used heavier horses, or at least more muscular horses (which isn't always reflected in the usual "height-only" measurements), and that included early heavy chariots, which are specifically noted in a Mitanni manual as requiring 'big' horses.
Light Cavalry - which usually means unarmored or More Lightly Armored than Heavy Cavalry, could use smaller or less muscular, frequently bred for speed horses. On the other hand, light cavalry did not necessarily require less training for man and horse, and therefore might be almost as expensive to Maintain as heavy cavalry - Roman manuals describe very elaborate training facilities ad exercises for their 'light' javelin and sword-armed cavalry that would not be cheap to build and keep in near-constant use.
People who are not expected to fight on horseback always get the 'leftover' horses. Whether they are mounted infantry or Dragoons, or Peter the Great's Corps Volante ("Flying Column"), their mounts tended to be whatever was available on four legs and We Thing It Was Alive This Morning.
The heavy/muscular and well-trained mounts for Heavy cavalry were always expensive, both in the training and growing time (no European Army from 17th to 20th centuries, when there are good records, would accept any horse less than 5 - 6 years old, so someone has to feed those critters for those years before you can get any use out of them) and in the feeding: every horse requires acreage for pasture, and frequently 'extra' food like oats/fodder to supplement the pastures, especially in winter. This costs money to maintain, and takes away acreage that could feed people.
Lighter and less-trained horses, if the pasture acreage is available, can largely feed themselves - witness the herds of 'wild' horses in various parts of the world even today.
And, if Heavy Chariots required Heavy Horses, the average Modern Era 'light' howitzer weighs about 5 times more than any chariot and every army using horses in quantity (specifically, the German and Soviet armies in WWII) recently had a special classification for the largest horses: "Draft" meaning suitable to haul heavy wagons and guns. In the 18th century, the average Field Gun/Artillery (French 8-lber, for instance) required a 6-horse team to haul the gun, and 4 - 6 horse teams for each caisson carrying ammunition for the gun - a total of 15 - 30 or more Draft Horses per gun. In every European army, the artillery used up far more 'heavy' horses than the cavalry did.
For Reference: a 'Hand' is 4 inches, or 10.16 centimeters. The term has been used since ancient Egypt, when a Hand was 5 'Digits' (fingers), or about 94 millimeters. The 4 inch Hand was standardized in England in 1540 CE by Henry VIII, and by International Inch standards in 1959.
A Stone is 14 pounds, or 6.35 kilograms, and is now used strictly in the United Kingdom to measure body weight, although back in the late Renaissance 'measuring stones' of more or less standard weights ranging from 5 to 40 pounds were used to weigh everything in balance scales all over northern Europe. The Russian Army for a while even measured their howitzers by a 'pud' that was roughly 16 pounds each: the standard field howitzers were 1 Pud and 1/2 Pud.
