UPDATED
This could be a coincidence, or maybe close to the answer but not quite, but it is interesting that...
All of the Salon Words anagram to a word of French origin
- except for taipan, which anagrams to patina, which is of Italian origin, or pinata, of Spanish origin
Whereas for Not Salon Words:
Only 'dasani' and 'buyer' anagram to anything; 'naiads' which is of Greek origin and 'rebuy' which is old English. The rest don't seem to anagram at all (subject to choice of dictionary).
So perhaps
A Salon word is a word that anagrams to another English word of French or Italian (or Spanish if we use Pinata instead of Patina) origin. Perhaps in linguistic terms these are a common group.
However, you may have noticed that most of these words
Can be written with an accent mark/diacritic, for example naïve, piñata, café, etc. I originally considered this, but ruled it out with 'rose' which WAS of French origin, but old French rose seemed to have no diacritic.
Upon further consideration
The French rose derives from old English rōse, and it seems in some French there is even rosé. Further, SORE could anagram to EROS which comes from the Greek erōs.
So in the end it seems possible that is a salon word is one which
anagrams to a word which, in the process of forming modern English, retained its exact spelling but the diacritic was "trimmed off" as in, got a haircut at a salon.
With many of these words, perhaps all? it is even still considered proper English to retain the diacritic, but here it is simply missing, although I am not sure if that is relevant or a more correct answer. For example, perhaps all Salon words also have an English equivalent that traditionally had a diacritic, but again I'm stuck at sore. There may be a word in modern English with a diacritic that it anagrams to, but am not sure.
Full list:
face > cafe
evian > naive
mater > armet
said > dais
taipan > pinata
coupons > soupcon
sore > rose or eros
rentee > entree
