Kindersley (1997) states (in the Tonga page) that the flag with the red cross coupee was adopted in 1863.
The plain Red Cross or Red Crescent flag is primarily used to identify medical units of military forces as provided by the 1864 and subsequent Geneva conventions, although national governments may authorize Red Cross or Red Crescent societies to use the flags provided there is no possibility of confusion with the praimary use of the flag. The convention also allows the various international Red Cross organizations (ICRC and IFRC) to use the symbols. Therefore, the plain Red Cross is not a unique identifier of the ICRC.
The basic right to use the Red Cross and Red Crescent flags does not derive from either of these organizations. It is an inherent right of all states that are parties to the Geneva Convention of 1949 and any other belligerent that might be engaged in combat in accordance with the laws of armed conflict.
Vexillologist Léon Nyssen received a letter (in French) from ICRC quoting the Acts of Geneva Convention of 1949:
"C'est de propos délibéré que l'on n'a pas voulu fixer la forme de la croix rouge, ce qui eut ouvert la porte à des abus dangereux... Si la forme de la croix avait été fixée de façon immuable, n'aurait-on pas cherché à justifier des attaques contre les bâtiments protégés par la convention en prétextant que les signes n'avaient pas les proportions prescrites?..."
A tentative translation:
"On purpose it has been desired not to determine the shape of the red cross, which determination would have given way to dangerous abuses...If the shape of the cross had been determined in an immutable way, would not one try to justify attacks against buildings protected by the Convention, arguing that the symbols did not have not the settled proportions?...
Nowadays it is normal that the Red Cross symbol in ordered (i.e. peaceful) conditions is modeled after the Swiss flag, more or less. However, looking at old documentaries it seems to me that before, say 1950's, the shape of the cross was usually much thinner (even to one third of the width of the Swiss cross).