lSpace Fréchet manifolds as diffeological spaces (changes)

At the start of section 4, the article says (emphasis added):

By Fréchet manifolds we mean a $C^\infty$ manifold modeled on Fréchet space, which is defined in the standard manner by means of any definition of differentiability, because all of them are equivalent, as it has been demonstrated in [section] 3.

But in the Historical remarks at the end of chapter 1 of a convenient setting for global analysis?, it is said that there are 3 inequivalent notions of “infinitely differentiable” for Fréchet spaces.

Digging a little deeper, my guess is that the different types of differentiability referred to in this article are all variants of the idea that a function $f \colon E \to F$ is differentiable if it is directionally differentiable, i.e., all directional derivatives exist. The variants are presumably related to how continuous the resulting directional derivatives are in both variables. Checking this will involve tracking down the references.