To be honest, it was neither of your scenarios. The United States did not play a single decisive role in the First World War. You are right when you say that the British and the French did most of the heavy lifting on the Western Front (when I say British I am if course also referring to the Commonwealth forces of Australia, New Zealand and Canada) and you are also right when you say the by 1918 the French and the British were becoming increasingly desperate. Four years of heavy fighting had taken its toll on French and British manpower. Britain was being forced to disband understrength battalions to reinforce other understrength battalions (as was Australia and Canada) and the French were only just regaining their offensive capability after the mutinies that had reduced the French army to a largely defensive role in 1917.

That said, Germany was also desperate. Despite the capitulation of Russia in 1917 and the freeing up of German forces on the Eastern Front, Germany was also acutely aware of the looming arrival of US forces in France. Britain and France were drained in terms of manpower and Germany was at breaking point. The German High Command knew that the only chance it had of achieving victory was to launch a major offensive and achieve an overwhelming victory before US forces arrived in large numbers. Thus was born the Spring Offensive of 1918.

Although overwhelmingly successful tactically, the Spring Offensive lacked a realistic and clear objective and was subject to changes determined by events. By the time it petered out in mid 1918, the German army had squandered its manpower advantage and had failed to achieve the knockout victory it required. The Allies on the other hand, though reeling from the shock of the German successes, were planning to go on the offensive. Even before the Spring Offensive had concluded, the French army launched a major counter attack, the Battle of Soissons. This attack, conducted by 24 French divisions, 2 US divisions under French command and 8 under US command. The attack drove the Germans back some 5 miles and saw almost 30,000 prisoners taken. The role of US troops, along with Italian and British divisions, was a supplemental one with most of the fighting being conducted by the French.

The largest and possible most important contribution made by the US during the First World War was during the Meuse-Argonne Offensive where 2 US armies launched attacks on german positions on 26 September. This was in conjunction with French armies of which there were 8. During this time, there were 3 British armies, the First, Third and Fourth Armies, fighting in the north of France and in Belgium.

While the United States contributed large numbers of men to the Allied war effort in France and whist its formation fought in some important engagements, the American Expeditionary Force never achieved the decisive role that it would take in France and Germany in 1944-45. In 1918, The majority of the fighting was still being done by British and French armies with US forces supplementing them.

Now, that said, the United States still played a critical role in the Allied victory, that of banker. The United States effectively bankrolled the British and French war efforts, issuing loans and bonds throughout the war, allowing the Allies to continue paying their soldiers, to buy materiel (much of which was being manufactured in the United States) and to feed their populations. Without this financial support, the Allies would have been hard pressed to continue their war effort.