(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a monomorphism of $R$-modules, then the induced morphism $M'\otimes_RN\rightarrow M\otimes_RN$ is a monomorphism, too. One can show that an $R$-module $N$ is flat (i.e., $M$-flat for every $M$) if and only if it is $R$-flat.
Let's call an $R$-module $M$ self-flat if it is $M$-flat, i.e. if whenever $M'\rightarrow M$ is a monomorphism of $R$-modules, then the induced morphism $M'\otimes_RM\rightarrow M\otimes_RM$ is a monomorphism, too.
In the aforementioned question examples were given that show that not every module is self-flat. On the other hand, it is readily checked that flat modules, injective modules, monogeneous modules, and semisimple modules are self-flat.
What are other classes of self-flat modules?
(A quick search among the usual suspects (Lazard's Autour de la platitude, Bourbaki's AC, Lam's Lectures on rings and modules, etc.) did not turn up anything about this notion.)