The fact is, the word “Maya” should be used in all three of the above situations. Here are some specific examples showing correct usage:
As a singular noun: He is (a) Maya.
As a plural noun: They are Maya.
As an adjective: Maya astronomy, Maya calendar, Maya civilization, Maya clothing, Maya culture, Maya folklore, Maya food, Maya glyphs, Maya hairstyle, Maya heritage, Maya hieroglyphs, Maya pyramids, Maya religion, Maya ritual, Maya warfare.
The name of the spoken language: this is a little bit more complicated. “Mayan” refers to the family of 29 to 32 languages spoken by the Maya. However, there is one exception to this: according to The Open School of Ethnography and Anthropology, “The use of these words suddenly gets complicated…There is one Mayan language whose proper name is “Maya”! The native speakers of Yucatec Maya call their language Maya. Maya is the correct proper name of the language that they speak.”
So, at Yucatan Today, because we are usually referring to Yucatec Maya when we refer to the language spoken here, we will continue to call this language “Maya”.
More information: osea-cite.org/program/maya_or_mayans.php