Cavalry's not really my area of expertise, but as someone who studies medieval warfare it is something I'm required to know a few things about (most of the horse related stuff I study relates to transport and logistics). This past July I was attending the International Medieval Congress in Leeds and I decided to attend a session on horses and medieval cavalry to better my own knowledge, and it is from that session that I'm basing my response. In particular, I'd point to the work of Jurg Gassmann, an independent scholar and reenactor specialising in medieval cavalry who was communicating not only his own work but that of several other scholars whose names I totally missed because it was an early morning session and I was tired. You can check out some of his work on his academia.edu page. I would also stress that my comment only applies to the central and later Early Middle Ages. Cavalry has been used in warfare for a long time and how it was used varies between regions and eras, my answer is by no means applicable to all of those.

In his paper on medieval cavalry, Jurg stressed that the wedge shape would have been the primary formation for medieval cavalry, although with the caveat that the really tight perfectly shaped wedge you see in Total War games isn't totally right. In practice, the lead rider might be a line of a couple of leaders stretching further in front of the wedge. The reason for this shape is that it's much more manoeuvrable. If you think about it practically, if everyone is in an equal line, who steers the unit as it approaches the enemy line? Battles are rarely fought between perfectly formed lines on flat open fields, some movement is required to position yourself before your charge. If you're in the middle of a line, you can't really see if the unit is turning until the person next to you starts to turn, which makes it very unwieldy and restrictive. In a wedge shape, the person at the front of the wedge is free to ride in any direction to suit the changing needs of the battle, and the wedge spreading out behind him allows most of the other cavalry soldiers a clear line of site to him to allow for quick response to his changing movements. Given that one of the greatest strengths of cavalry over infantry is greater speed and mobility, using a formation that supports this is only logical. It does present some greater risk to the person in the front when the lines inevitably clashed, but that's always a risk for whoever's up front in war.

I have no idea about diamond shapes, sorry.

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Edit: I totally just wrote central and later Middle Ages on instinct (when do I ever write about early?) but Gassmann's argument is actually most applicable to the Early Middle Ages.