What are your sources? I ask because most documents do not place the red cross on the Bauceans flag (the black over white) until much later than our period in history. There are some references to a somewhat smaller red cross in the center of the flag, or on tabbards or armor, but they appear to be somewhat later as well.
However, as far as I know, the flag you posted (with the large red Spanish-style cross) was first used in Portugal, LONG after our period of history.
The shield you posted is clearly French, not Templar order - it is marked with the "fleur-de-lis", symbol of the French monarchy, and so would not have been allowed by orders such as the Templars or Hospitallers.
The only existing art I know of that shows a Templar shield in correct period is a painting of two guys on a horse, and their shields are black on the top third, and white on the lower 2/3, with no other markings.
The Rule of the Templars, as specified by the Council of Troyes, 1129, specified that all in their order were to wear habits of a single color - white, black, or brown - and were specifically forbidden to decorate them with such things as the Crusader's Cross shown in your previous post.
Link to a copy of that in English:http://www.the-orb.net/encyclop/religion/monastic/t_rule.html
And so until proven otherwise, I and many other historians around here will consider your sources of information to be highly questionable at best. Sorry about that ... but there's just a lot of really bad information going around about this period in history.
The mod uses black or white tabbard with the small red cross on the upper left side of the chest - probably a little later period than it should be, but necessary to mark the Templars. Templar helmets and shields are black and white. There are two horse barding models, one black and white, and the other with a red cross on the white portion. Again, the red cross may be later period than intended, but it serves well to mark the Templars, so accomodation has been made. That is as of the .808 developers' version (which was not released for technical reasons) - and is not necessarily fixed for future versions.
The solid white with the big red cross in the middle is just plain wrong. It's a French Crusader's Cross pattern, not Templar order. Easy to see how they got mixed up, but not a mistake we care to replicate.