The Persians, Arabs (at least after they'd already taken over most of Middle East) and the Byzantines actually used relatively similar cavalry techniques. Small wonder, as they were constantly butting heads in similar conditions and on ultimately fairly similar resource bases.

The Turks and other nomads had a rather different approach, but this is hardly anything new. Steppe nomads have a quite different recruitement and ecological base than sedentary peoples. Although they too tried to get as much armoured cavalry to the field as they could muster.

Anyway, the horse-archer was a staple among all Eastern cavalry. The nomads tended to employ them as loose-order skirmishers, the sedentary peoples who lacked the vast horse herds needed to support that tactic as close-order multipurpose cavalry instead and preferred to shoot standing. The latter were also typically able to armour their soldiers rather better and give them more thorough hand-to-hand combat training, which made theirs a by far rather more all-purpose troop type than the light steppe skirmishers.

One gets the impression that the difference between the horse-archers and more dedicated heavy shock cavalry among the sedentary nations tended to be blurry indeed, and the types were in practice often one and the same. Heck, the Mamluk horse-archers were trained to use their bows as close assault weapons, and probably weren't the only ones...

Anyway, as a side effect of the fairly extreme proliferation of composite bows and archery in the region since ancient times heavy armour was popular - the massively armoured cataphract developed among the archery-crazy Central Asians and spread rapidly in all directions, remember ? Being able to outlast the other guy in an archery fight by virtue of sheer resiliency, or ride through his missiles with impunity, is an obviously useful ability for soldiers.