The changing of the scene/method of the killing is to be expected in such a widespread ballad. In some of the more central/possibly earlier international versions the Knight gives the girl the option of being killed by his sword, hung or drowned, so the suggestion of drowning was already implanted. If I remember correctly it's the French and Polish versions that move towards the drowning method, and therefore most likely the source of the English ballad.

Whatever, it is the girl's reversal that is the crux of the story. The method is not as important (IMO).

What Child's (Grundtvig's) synopsis of the continental versions doesn't tell us is whether as the ballad moved from language to language did the basic story just pass on or were there direct translations of ballad to ballad. The probable answer is both. I look forward to seeing if Kemppinen throws any light on this.