I believe everyone knows "in a local Noetherian local commutative ring, a f.g. module is free if and only if it is flat".
The proof is in the question flat f.g. modules over a commutative, local, Noetherian ring are free . We only need to notice that $k \otimes F$ anf $k\otimes M$ are vectors space of the same finite dimension.
Anyway, I do not see where the Noetherian property is used. NAK lemma does not require it.
Thank you for your help.