Spotting it
The reader flags bIII on one chord: a major chord whose root sits a minor third above the tonic, in a key whose own tonic chord is major, with at least one note outside that key. In C major the chord is Eb major, spelled Eb, G and Bb. Only G belongs to C major; Eb and Bb are the two notes the fit column names.
In the table
The numeral is bIII: upper case for the major triad, and a flat because the root is a semitone below the key's own third degree. In C Eb F C the chord sits between the tonic and IV, and it is the one row with anything outside the key. The line ties between C major and C mixolydian, and neither scale holds Eb, so the move is flagged whichever of the two you read it in.
Borrowed before anything else
Eb major is also a chord a third away from C with the tonic's own quality, which is the shape of a chromatic mediant. The reader checks borrowing first: every note of Eb major belongs to C minor, the parallel key on the same tonic, so in C Eb C the chord is read as bIII and the chromatic mediant rule stays silent. That is the order the source for the chromatic mediant gives: rule out borrowing, then a chord leaning on the next one, and only then call it a mediant.
What is not flagged
In a minor key the same chord is part of the key. Cm Eb Cm reads in C minor, the numeral is still written bIII because every numeral here is counted from the major scale, but nothing is outside the key and the move does not fire.