The alternative derivation of the flag is that it comes from one of the flag designs associated with St Denis. There are earlier examples of its use in illustrations from (I think) 14C or earlier, but not with any specific designation or ownership.
All of the vielles bandes had the same design with a different background colour and the white cross was a feature on the vast majority of French infantry flags pretty well from their creation to the revolution.
A simple cross on a flag (in various colour combinations) almost certainly derives from the Crusades and it is a very common feature on military flags throughout western Europe so I don't think it is out of the question that there may be more than one 'original' that might have inspired the flag.
It seems that many banners and devices from pre-heraldic times survived into the heraldic period but not always associated with exactly the same region or persons.
As to your final query, I have no more idea than you but I used the white X on Red for no other reasons than it was a flag known to have seen earlier military use and it was what they ended up with. I suspect fancier designs, possibly with devices of their commander or even Royal ones, were used but I know of no evidence to back that up.