Okay, who used to, or still does, add that after, “You would even say it glows”?
Yes. I also mentally add in “Like Monopoly!” after the reindeer games line.
I remember a whole series of these after a whole bunch of lines (e.g. “Santa came to say…” “Ho ho!” and “As they shouted out with glee…” “Whoopee!”).
Bart Simpson, at the end of Simpsons Roasting on an Open Fire.
MilliCal learned all these Rocky Horror Show-esque interjections at school, and uses them all the time.
Kids these days.
“…used to laugh and call him names” Like Pinocchio
“go down in history…” **Like George Washington.
**
Though with the phrase “go down” and “history” we should change it to "like Lewinsky!"
I just usually say “GLEEEEEEEE!” in an annoyingly cheerful voice
that actually scans:
“You’ll go down like Lew-In-Sky!”
“all of the other reindeer…” like Cupid (I’m not sure why Cupid go the rap for being the ringleader of the teasing.)
“used to laugh and call him names…” like stupid
LOL. Oh, duh. Now I see why it had it had to be Cupid. 
We always used to end with “You’ll go down in history…” like a light bulb!
I’m the only one who’s made up an X-Men parody, right? “Cyclops the red-eyed mutant…”
Right. I’ll shuffle off now.
you all realize, of course, that this song mentions the 10th reindeer
wait for it
“Olive, the ***other * ** Reindeer”
Oh, dear sweet Pete, if I hear that “joke” one more time… I’ll lose it.
Not being big fans of ol’ Georgie up here, our line was “You’ll go down in history.” “Like Columbus did!”.
Interesting how many different ones there are. From when I was in High School (New Orleans area, mid-80s):
reindeer (reindeer)
shiny nose (like a lightbulb)
saw it (saw it)
glows (like a lightbulb)
reindeer (reindeer)
names (like Mudface)
Rudolph (Rudolph)
games (like football)
Santa came to say (Hey Rudolph!)
loved him (loved him)
glee (GLEE!)
reindeer (reindeer)
history (like Columbus)
Me: …and Led Zeppelin wrote a song about her!
Mr. Rilch: … 
Me: C’mon; it’s one of your favorites! Olive my love, Olive my love, oh Olive my love to youuuuuuu…
I believe the Simpsons version that Floyd mentioned went thusly:
…say it glows (like a lightbulb!)
…call him names (like schnozilla!)
…reindeer games (like strip poker!)
…you’ll go down in history (like Attila the Huuuun! [sung like “good evening frieeeends”])
Homer got progressively madder with Bart’s additions, and, of course, ended up choking him.
The Simpsons version also featured a botched solo by Homer.
“Rudolph, get your nose over here
And help me guide my sleigh…today”
Doesn’t add anything to the discussion, but I always found it funny.
Our Rudolph went down in history “lik Atilla the Huuuuuuuuun!”
As a kid it was always: light bulb, Monopoly, Pinnochio, Micky Mouse (after the history).
I grew up near Atlanta and this is how we sang it:
reindeer (reindeer)
nose (like a light bulb)
saw it (saw it)
it glows (like a flashlight)
reindeer (reindeer)
names (like Pinnochio)
Rudolf (Rudolf)
reindeer games (like Monopoly)
<snip>
loved him (loved him)
with glee (yipee)
reindeer (reindeer)
in history (like George Washington)
…
Why would reindeer play Monopoly?
Because they were missing some of the pieces out of the Risk box