For example, take a C chord. One of the requirements for a chord to be a chromatic mediant is that it is a major/minor third away from the root. So why are chords like Cm, Fm, C♯ and F♯ considered chromatic mediants as well?
Your understanding of a chromatic mediant relationship is correct, so the chords you listed do not of have a chromatic mediant relationship to C major.
However, those four chords (Cm, Fm, C♯, F♯) all have roots a third away from A. A minor is the relative minor of C major, which means the two keys share the same key signature. So I think the source you were reading was listing out the chromatic mediants for the relative minor.