Spotting it
The rule looks at one chord. Its root sits a minor sixth above the tonic, its triad is major, at least one of its notes lies outside the key, and the key's own tonic chord is major. In C major that chord is Ab major, spelled Ab, C and Eb, and two of its three notes are outside the key, so the fit column names both Ab and Eb.
In the table
The numeral on its row is bVI. In C Ab C the chord sits alone between two tonic chords. In C Bb Ab Bb C it sits between two Bb chords, and that line reads in C mixolydian rather than C major, because the Bb chords fit the mixolydian scale; the Ab still has two notes outside it, and the move is flagged there too. The same line also fires the three-chord move bVI–bVII–I on its last three chords.
Not a chromatic mediant here
Ab major a third from C major has the tonic's own quality, and the source for the chromatic mediant lists it among the four chords of that kind in C. It also notes that the chord is a borrowed one, and the reader settles the overlap by checking borrowing first: every note of Ab major belongs to C minor, the parallel key, so the chord is read as the flat six and the chromatic mediant rule does not fire.
What is not flagged
In a minor key the chord on the lowered sixth is part of the key. Am F G Am reads in A minor with F as bVI and nothing outside, and the move stays silent, because its rule only applies where the tonic chord is major.