What you seem to be describing is the flag of the Red Cross (ICRC)
Perhaps I missed something? Is there some sort of Templar connection with the ICRC?
I thought the Swiss flag was originally the flag of the Canton of Schwyz, developed in the 1200's.
But maybe I wrong?
"Perhaps I missed something? Is there some sort of Templar connection with the ICRC"
Oddly, yes. The Red Cross was founded by a York Rite Freemason named Jean Henri Dunant about 1860 or so.
(The York Rite is the expressly Christian-only branch of the fraternity.)
The current state of the organization would have the man (a lay preacher) spinning in his grave.