Arthur Lee’s Love too, in many songs.
But those songs really aren’t about the bands. In fact, I always kinda had the idea that the bands named themselves after the songs.
KMFDM has made it a recurring thing to the point of it being a joke now.
I love that album. Also, fun fact, the lyrics to “Drivin’ Sister” name check one of their earlier songs, “Half Moon Bay.”
They name dropped the name of each member, too, in the song Ramona:
Hey, Johnny
Hey, Dee Dee
Little Tom and Joey
You know we’re goin’ over
Sweet, sweet little Ramona
The Court of the Crimson King by… King Crimson. Boy, they screwed that up. 
One version of Hyena Stomp by Jelly Roll Morton features the words “Jelly that’s terrible” spoken right after the beginning of the song. One of the song’s gimmicks is that the band “messes up,” stops, and starts over, followed by laughter. It was recorded in 1927.
That song was by Motörhead, though.
In the self-reference dept, though obviously not first, its hard to beat:
“Talk Talk” on the album Talk Talk by Talk Talk
Living in a Box (a mostly-forgotten British band from the 80s) did that as well. Band name-album title-single title.
Solomon drops his own name in Song of Solomon. Get earlier than that 
Same year Ma Rainey recorded “Ma Rainey’s Black Bottom”.
Oh, we have a horse race folks! Wonder if we can get back to Stephen Foster name checking himself in one his pieces! 
Ladies and gentlement, I give you La déploration de la mort de Johannes Ockeghem (1497 or shortly thereafter) by Josquin des Prez:
Nymphs of the wood, goddesses of the springs,
Skilled singers of all nations,
Change your voices, clear and lofty,
Into sharp cries and lamentations,
For Atropos, terrible satrap,
Has inescapably ensnared your Ockeghem,
Music’s true treasure and master,
Who henceforth no longer escapes death,
Of whom it is a great sorrow that the earth must cover him.
Put on the clothes of mourning,
Josquin, Brumel, Pierchon, Compere,
And weep great tears from your eyes,
For you have lost your good father.
May he rest in peace.
Amen
Johannes Ockeghem was a renowned composer of the period; Josquin and the others listed in that line were the leading composers of the next generation.
Beat that!
He’s referred to himself as “Killer” in several songs, but in the studio version of “Rockin’ My Life Away” he sings:
“My name’s Jerry Lee Lewis and I’m dern (damn?) sure here to stay” (around 2:38)
“Yellow Balloon” by the Yellow Balloon. The flip side of the single was “Noollab Wolley,” so that should count, too.
Now we’re talking! I had a quick look through stuff a good bit after that era (1700s), but I wasn’t sure even how to refine my searches to keep them from being anything but needle-in-haystack attempts. It seems to me that the idea of inserting oneself into a song must date back quite a ways. Like in storytelling songs that involve yourself, if you are singing about other characters and what they say to you, it would seem sticking your name in the lyric would be a natural impulse.
Thorogood name checks himself in the same song-
“Now Arlene took a-me by my hand
She said lonesome George you don’t understand”
I had completely forgotten about that wonderful work, perhaps my favourite from Josquin.
The way voices and names softly overlap in the relevant passage is magical (Josquin: Nymphes des bois - Graindelavoix - YouTube).
Metal Church’s debut album was called Metal Church. The second song on this album is called Metal Church, and “Metal Church” is mentioned four times in the chorus.
They really wanted you to know who they were. 