What is the intuition behind the name "Flat modules"?

I am studying Atiyah and MacDonald's book "Introduction to Commutative Algebra" and I have just read the definition of a flat module.

It seems to me that if they have called that kind of modules "flat" there must be some sort of geometric intuition or something like that. Why are they called so? Or is it just a meaningless name?

Thanks

Flat modules were first defined by Serre in his seminal paper Géometrie Algébrique et Géométrie Analytique (GAGA). (For an English translation, see this link) (or this direct link). In that paper, Serre gives no motivation for the name. I actually asked my commutative algebra professor your same question -- why do we call flat modules "flat"? -- and he told me that Serre didn't have a good reason for it. I don't think the name has geometric significance.