Mary then quits her job and steals the company's records, and mails the two former clients their records from Lacuna.
The film returns to the present, after Joel and Clementine have met aboard the train. They both come upon their Lacuna records later that day, and react with shock and bewilderment, because they have no clear memory of having known each other, let alone having had a relationship and having had their memories erased. Joel beckons Clementine to start over.
This above excerpt is from the wikipedia page https://en.wikipedia.org/wiki/Eternal_Sunshine_of_the_Spotless_Mind#Plot
There is no mention of this scene in IMDB but if you see the scenes its a mark of continuity for the audience to perceive the order of events. The mark is thus, from the fall on the hard frozen surface of the lake, Clementine took Joel after they met for the second time afresh in the train.
So you are correct on guessing which fall resulted in that wound or mark.