Diatonic means two tonics or root notes. It refers directly to the duality of the Major scale and its relative minor scale
when talking about chords, Diatonic means found within the scale and includes I-I(#5)-ii-iii-III-IV-V-#viib5-vi-viib5
in the C major/A minor diatonic scale these would be
C-C(#5)-Dm-Em-E-F-G-G#mb5-Am-Bmb5
the chords from C major are
C-Dm-Em-F-G-Am-Bmb5
the chords from A minor are
Am-Bmb5-C(#5)-Dm-E-F-G#mb5
Other chords
He probably used the phrase Chromatic chords or borrowed chords
These include the chords already found in the diatonic scale as well as borrowings from the parallel minors and majors and other chromatic alterations. The Neapolitan chord is actually a Neapolitan 6th which is chord bII in 1st inversion within the diatonic context , in C major this would be Db major with an F in the bass and it would be used as a substitute for chord IV or chord ii leading to chord V.
There are several other 6th chords available including the German, Italian, French and the so-called Australian 6ths: all four of these are chromatically altered dominant 7th chords but render the 7th as a #6th (hence the name) there are very specific uses and resolution for these European 6ths which fall outside of any useful discussion here
Your chords
I actually get 6 chords:
G#-B-D# ~ C#-D#-G# ~ A-C#-E ~ A-B-E ~ E-G#-B ~ G#-B-D#
G#m - G#Sus4 - A - ESus4 - E - G#m
note that the two sus4 chords are in 1st inversion (the suspended 4th is in the bass)