Etymology of the French game name dames

 I elucidate my method by the present-day verb damer meaning to tamp down ground. The first vocal of damer is pronounced with the clear AA as in French dame. The English sound system has no words with the clear AA, as far as I know as a speaker of Dutch. An older form of the verb is dammer, pronounced with what I call the dull A, like in English words as calm and past, but shorter.
The verb dammer is a derivative of the word dam, dike, dam to stop the water.
The development from dammer to damer fits within a pattern. Medieval French had many words with the dull A, but in due course this dull A was replaced by the clear AA. This leads me to my research question: was the game name dames one of these words?
And this question brings me to a clear research target: seek references of the French game name dames and derivatives. If the words I found always had the clear AA, the etymon of the game name is dame. In this case the literal meaning of jeu de dames is “game played by woman of high social standing”. If the words I find had the dull A, the etymon of the game name is dam. In that case, the game name dames is one of the many words where French speakers replaced the dull A by the clear AA.
One of the research methods my teachers supplied is to look for double forms. Suppose that the game name dames is one of the words where the French replaced the origin dull A by the clear AA. In such a case there is always a transition period where one group of speakers hold on to the old name dammes and another group that switched to the new form dames. In this case I can perhaps find double forms. And I found them indeed. Not in French however but in Flemish: the French exported their doubt about the right pronouncement to Dutch speaking regions! I quote three Flemish lexicographers:
Corderius (1552): Laet ons dammen. De schijven om te damen
Sasbout (1576): Dammen, met damen spelen
Kilianus (1588 en 1599): Damen, dammen = ludere scrupis duodecim.
In a later correction, Cornelis Kiliaan ‒Kilianus is the Latin form of his name‒ crossed out the form dames in his personal dictionary, proof that in Flemish the new form damen did not succeed in ousting the older form dammen. Also a proof that the verb dammen hold a strong position, i.e. that it was frequently used.
There are more proofs that the etymon of the game name dames was the word dam, but I confine myself to these three lexicographers.
Why the promotion row named after a dike? Because the march of the singleton stops at the row of squares bordering on the edge of the board, could that be the association?
For the case of completeness, I give the etymology: dam = dam, dike > dam = raised edge of the draughts board > dam = promotion row, row bordering on the raised edge of the board where the singleton is promoted to doubleton > (jeu de) dames = “game of the promotion rows”.