Mae Mai MT~
It means 'mother branch' of Muay Thai. All of the basic and standard techniques of Muay Thai, for example, roundhouse kicks, standard punches, etc etc.
Luk Mai MT~
Means the more advanced and subtle 'tricks' of MT. For example, spinning elbows, hanuman tawai waen, pra ram jieb longka (Rama steps on the earth), etc etc.
It's not a style in itself.
Plus, don't learn from a non-thai teacher. They don't know it either, like your friend there has no clue whatsoever.
Note that many Luk Mai doesn't work because of the standard MT guard used nowadays, which is totally different from the elongated guard used 60 years ago which made Luk Mai possible.
I got two fists.. Don't make me use my head as well!