This one’s a little weird. They actually went by any number of outlandish names, such as The Inalienable Right to Eat Fred Astaire’s Asshole, the Vodka Family Winstons, and Ashtray Babyheads. At their first paid gig, the emcee forgot their name, and used the song title off the set list as their name. It stuck.
So, it’s a little like a chicken and egg problem.
DChord: I wanted to thank you for supplying an additional comedic depth to this 39 year old National Lampoon bit. I always thought it was funny that “Gregory Puck” might be having a fever dream, but his last line from the Ritter song is now the icing on the cake. Kudos! 
We now return you to your regularly scheduled broadcast.
Mr. Mojo Risin’
The Fifth Dimension in Aquarius/Let The Sunshine In.
James Taylor - Sweet Baby James
Probably an obscure one: “Bruce” by Rick Springfield. A lament about the similarity between his name and Bruce Springsteen’s.
She called me Bruce, Bruce
I can hear her calling Bruce, Bruce
I can hear her calling Bruce, Bruce
I can hear her
My name is Rick I’m gonna stick it to ya babe
How about…
“What’s Your Favourite Colour?” by Living Colour
“It’ll Chew You Up and Spit You Out” by Concrete Blonde
“My Name is Prince” by… er, I forget who…
Johann Sebastien Bach – The Art of Fugue
I’d forgotten about that. That pretty much wins the thread.
ETA: other than it not being rock, that is.
Joe Walsh, I Can Play That Rock & Roll
If you, if you wanna party at the next election
Only one way to go
Put on a rocking rock and roll selection
Turn up and vote
And you can check out anytime you want
Just call me Joe
And I can play that rock and roll for ya…
Metal band Manowar namechecks themselves quite often.
Hank Williams, Jr. mentions himself by name in more than one song.
Wille Nelson & Waylon Jennings in Luckenbach, Texas:
Let’s go to Luckenbach, Texas, Willie and Waylon and the boys…
Although they apparently didn’t write the song themselves.
“Creeque Alley” is a good example of an autobiographical song, which includes, but goes beyond, mere name-checking. It does, as you said, mention all members of the group, as well as some of their contemporaries.
Another such autobiographical song is The Kinks’ “The Road”:
In the song Ramona by The Ramones, Joey sings
Band name: Metal Church
Name of first album by Metal Church: Metal Church
Name of second song on Metal Church, by Metal Church: Metal Church
And in the song Metal Church, on Metal Church, by Metal Church, they say,
What do I win? 
You win a clap and a from me, because I forgot to list that song and back in the '80s they were one of my favorite bands.
But since I already listed Talk Talk by Talk Talk from the album Talk Talk (which contains the phrase “talk talk” more than 20 times, that’s all you get.
Yeah, I think he likes to name-check himself a lot. Here’s the start of “Jukebox”:
“…I’ll fake it
But some how-a old Jerry comes through
Sometimes I feel like I, I just can’t take it
But somehow the Killer comes through”
And in his version of “Honey Hush” he sings:
“I know you, pretty baby
You ain’t nothing but an alley cat
The Killer knows you, honey
Woman, you know what the Killer is talkin’ about”
He also name-checks himself and a couple of his fellow Eagles in “Shut Up.”
Hey Joe, how ya doin’, how’s your album, how’s your tour
How’s Don Henley, how’s Glenn Frey,
and who, what where and when and why
They could not shut up (can’t shut up)
They couldn’t shut up (can’t shut up)
So I hung up (can’t shut up)
Would “Stray Cat Strut” by the Stray Cats count?
I don’t think this one’s been mentioned yet…
Dr. John, “Gris-Gris Gumbo Ya Ya”: “They call me Dr. John, known as the Night Tripper…”
“Killer Queen” by Queen.
“Who Are You?” by The Who.
(Those two may not quite be direct references to the band, but they are not mere coincidences either.)
“Down at the Doctors” by Dr. Feelgood. (The chorus contains the couplet: “Come on down to the doctors/ Make you feel good all night.”)
“Blockheads” by Ian Dury and the Blockheads.