Agreed, with the reservation that winning on the Marne doesn'tr necessarily end the war..
True enough. It would then come down to how the French view the future. Do they still think they can win in a reasonable period of time.
In the aftermath of the Marne many of the French seemed to think victory could be achieved fairly soon. They thought the Germans would be demoralized and the Russians were still in East Prussia with dreams of early victory.
This encouraged them to go on the offensive at the Aisne and they were willing to take casualties. Even after the defeat of the Russians and the stalemate by December the French still looked forward to a few more pushes to bring them victory. So they endured!
Would they feel like that after a defeat on the Marne, followed soon after by the news of the destruction of the Russian armies in East Prussia?
Would they think that the strategic position had become hopeless and that winning would be too much of an uphill struggle to even contemplate?
Also what if the BEF had got mauled too? Would the British (at so early a stage) commit to raising a whole new army for a long war.
Kitchener knew it would be a long war but the Britsh people did not. Would not a defeat on the Marne jolt the British people into drawing back from the abyss?
So you can still get a short war if the French and British think that the task will be too great. If they don't think that then you still have a long war.