I disagree. The Byzantine Empire largely incorporated barbaros mercenary cavalry, such as Hunnic\Scythian cavalry, who used the advent of the Stirrup. At the time largely unknown to western civilization. Indeed from my understanding, the Byzantines first use of their own effective, non-mercenary cavalry was a type of heavily armored horse archer. Which eventually evolved into the Cataphract. Cataphracts, from my understanding actually used a bow, spear (lance), and short-sword.
Also, the Byzantines conscripted, especially in the Belisarius times, many Persian\modern day Turkey; men. A majority of whom fought in important battles, and as soldiers in the Byzantines conquests. Syrians, Persian, Scythians, all these were famed for their cavalry, and all these were at one point used in Byzantine military movements.
From what I understand, popular fashion simply started to transform the image of the dress\gear of the cavalry as the Byzantine Empires views shifted to the East. So, my point is, the Byzantine cavalry in its early Empire and Greek militia\ soldiers in the later Empire would most likely have worn the head gear of the Roman times.
*cough*

Anyway, Kryten is going to copy\paste another version you say? The more the merrier

.
Really though, they all look good, just a matter of taste.