“Behalf” is a handy little word, and lyricists like it because because it rhymes with “laugh”:
I’d like to say a word in her behalf
Maria makes me laugh
That’s Oscar Hammerstein, in The Sound of Music.
When I see depressing creatures
With unprepossessing features,
I remind them on their own behalf
To think of celebrated heads of state
Or ‘specially great communicators …
Did they have brains or knowledge?
Don’t make me laugh!
And that’s “Popular”, obviously, from Wicked, by Stephen Schwartz.
The world stands still
On my behalf
And I find that I’m in love
With Lucy’s laugh
And that’s “Lucy’s Laugh”, lyric by Christopher Dimond, from the song cycle Homemade Fusion.
Of these three, only Hammerstein gets it right. “In her behalf” means “in her favour” or “for her benefit”. “On their own behalf” means while acting as their representative, and “on my behalf” means the world stood still because I was going to do it, but I got held up by traffic or something.
Schwartz and Dimond mean “for their/my benefit”, but of course, that doesn’t rhyme. I suppose you could argue that Galinda makes the error, not Schwartz, but I don’t buy it.