The 13th Floor Elevators certainly played and recorded all their music on acid and their 1966 album had the word "Psychedelic" in its title. They were preforming in Austin well before the album was released. Not to be pedantic, but the word was coined by Humphrey Osmond in 1957, so the claim is credible.
Many authorities on the subject (e.g., Patrick Lundborg in 2010.
The Acid Archives (2nd ed.). Lysergia. p. 394)
assert that the Elevators are the "first" psychedelic rock group. Powell St. John (later of "Mother Earth") and a member of 13th Floor discusses "the life and times" here-->
http://www.musicliferadio.com/2010/10/024-sultan-of-psychedelia/.
Regardless, Roky Erikson was a very early casualty (and a sad one, too) of the "War On Drugs".