Rexing is what I (and my siblings) term speaking a song, instead of singing it - but doing so with such verve and panache that most people will not notice.
When done well, most of the audience will assume, will indeed recall, that the song has been sung, and perhaps sung well.
The name comes, of course, from the supreme and perhaps most extreme exponent of this way of performing a song: Rex Harrison. In the number above, from My Fair Lady - Rex speaks every word. I'm sure you will agree that he brings-off the feat marvellously; and few would feel in any way short-changed. A Master at work.
I don't think Rex was completely incapable of singing - he will sometimes throw in an occasional, strategically placed, fairly accurate - albeit brief - sung-syllable; as with perhaps his most famous (Oscar awarded!) number from Doctor Doolittle:
To be fair; even Rex could not get away with Rexing just any song - Nessun Dorma wouldn't be a good candidate, for instance. But if a song is Rexed with total confidence, it is remarkable just how much can be done with a very high degree of success.
(In Miss Saigon, Jonathan Pryce seems to have Rexed most of his dancing, as well as the singing - and received multiple awards.)
My brother was once doing the comic baritone role in an amateur Gilbert and Sullivan opera, and during performance week caught laryngitis. He all-but lost his voice - certainly could not sing a whole show; but there was no understudy. I asked him what would happen, and he stated his intention simply to Rex the whole thing! Which he did, all went well, and nobody seemed to notice...
(This might mean either that he Rexed-it superbly, or that his usual singing was not significantly different anyway... Make-up your own mind!)
But the plain fact is that most people, most of the time, can't tell the difference between Rexing and singing.
Therefore - if you are in doubt or despair about singing: Just Rex-it!