Turkmen vs Turki

Turkmen and Turki both are nouns.

Word	Noun	Adjective	Verb	Adverb
Turkmen	Yes	Yes	No	No
Turki	Yes	No	No	No

As nouns, Turki is a hypernym of Turkmen; that is, Turki is a word with a broader meaning than Turkmen:

Other hypernyms of Turkmen include Turkic, Turkic language, Turko-Tatar.

As nouns, Turki is a hypernym of Turkmen; that is, Turki is a word with a broader meaning than Turkmen:

Turkmen: a member of a Turkic people living in Turkmenistan and neighboring areas
Turki: any member of the peoples speaking a Turkic language

Turkmen (noun)	Turki (noun)
the Turkic language spoken by the Turkoman	a subfamily of Altaic languages
a republic in Asia to the east of the Caspian Sea and to the south of Kazakhstan and to the north of Iran; an Asian soviet from 1925 to 1991	any member of the peoples speaking a Turkic language
a member of a Turkic people living in Turkmenistan and neighboring areas

Turkmen (adjective)	Turki (adjective)
of or relating to or characteristic of Turkmenistan or its people or culture