Your graphical self online, borrowed from the word for a god descending to Earth in bodily form.
Documented
The descent of the god
Avatara in Sanskrit is descent, the crossing down of a deity into the world: the Bhagavata Purana lists Vishnu's ten, from fish to dwarf to Rama to Krishna. The word carried cosmic weight for two millennia before it carried a username.
Assessment: Documented in scripture and dictionary. The theology came first by a wide margin.
Stories
Ultima, Habitat, then blue giants
Games pulled the word down to Earth in the 1980s: Ultima IV named your character the Avatar in 1985, and Habitat in 1986 fixed it as your visible online body. The 2009 film made it blue. Each step traded altitude for resolution.
Assessment: Documented release history. The word's second career is fully archived.
Interesting claims
Every login is a small incarnation
The observation: the old concept fits eerily well, a higher being crossing into a lower world through a temporary form. The word did not change meaning; the worlds multiplied.
Assessment: A reading, not a record. But theologians and game designers arrived at the same noun independently, which is not nothing.