I can do better than asking my friend: since I made that post, I've tested it for myself. Not thoroughly, but I've been going to the same hair stylist for several years now. She usually does a very good job with the kind of haircut I prefer, but the first time I went after hearing that about moon sign, it came out not that great. The moon was in Cancer that day.
Next time I went, the moon was in Capricorn. My haircut came out the best ever.
Since then, I've managed to go back during Capricorn moons. Different times of year, so the Capricorn moon was in a different phase each time, and I still got a good haircut.
I don't know for sure if my hair stylist is a Capricorn, but based on this experience, it seems likely.
Maybe your stylist is a Cancer. If so, you should have better luck with them during a Cancer moon, or, if you can't manage that, a Pisces or Scorpio moon. If you go to a different stylist, then it depends on their sign.
My astrologer friend said the moon should be in the same sign or element as the stylist's sun, not their moon. And sun sign is easiest to find out, since pretty much everyone knows their sun sign--you don't even have to ask their sign, you can just ask them when their birthday is and find out that way--but only people who are very into astrology know their moon signs.
Perhaps some further evidence: a friend of mine who is a double Capricorn and wore her hair long for years got a sudden impulse to cut it short. She did it herself, just hacked it off in the shower, and it came out looking great. That was the day of the last new moon in Virgo.