The leading tone which is a half step below the Tonic is characteristic of music in a period called "Common Practice Period".
Modal music does not need to include the leading tone.
The hermonized chords in the order I gave at the beguining of the thread, correspond to the the fifth mode of the C harmonic minor scale. Where you shift the root to the fifth degree of the scale, therefore your home chord is not Cminor anymore; it is G major.
So the correct name for the mode is Phrygian Major , other possible name are:
Phrygian Major Third
Phrygian M3
Any of the following answers would be correct as well:
*G Jewish Ahaba Rabba.
*The fifth mode of the C Mohammedan Scale.
*The second mode of the F Roumanian Minor Scale.
*G Freygish Mode.