Muay - originated from the Cambodian/Khmer number for "one". if you know Khmer, you should know this.
Thai - is a recently new made up name for Thailand in which was adopted less then 100 years ago. They were originally called the Siam, from which the term Siamese cat was derived from
an ancient martial/warrior's art to protect the Cambodia/Khmer Empire and used as a fighting sport for entertainment, but now has been mislabel as muay thai as a marketing ploy to try give credit to Thailand instead of Cambodia (the Khmers).