Both the Greeks and later the Romans used the term Cataphract for some of their own heavy mounted troops (cataphraktoi / cataphracti ), but in the 4th Century B.C. the masters of armoured cavalry were employed within the Persian military. They still could not penetrate a cohesive phalanx, but otherwise they could sweep pretty much any other kind of troops before them.
The image to the left comes from the War and game blog (I’d like to know what the original source is actually). One can see why historians and history buffs refer to them as knights. Bronze helmet, bronze scales, the spear. Riders could carry a sword, a much longer spear, even a bow.
But what really set them apart was the horse bronze barding (a headpiece would often be used). It must have been a huge additional expense and a powerful symbol of status, but also effective protection. With both rider and horse protected by armor, a charge could be very hard to stop.
Heavy cavalry requires heavy support services: remounts, armor replacement, feed (outside the steppe anyway), possibly grooms. The whole system is backed by a breeding industry aiming to produce exceptionally sturdy large, aggressive horses.
No stirrups in 400 B.C. around the Mediterannean or Darius’ Persian Empire. They might have been used in India at the time (big toe only…), but there migrated West only later.