I was taught the song with Georgie.....we had to sing it in elementary school. I did Google and most are saying Henry. But did find a similar song about a jug with a hole and it has Georgie.
Yes, elementary school, exactly. Hmm, a Georgie and a jug, eh? Well, Georgie can suck his jug and Henry needs a boot in the arse, for not fixing Liza's hole. (in her bucket...get your mind outta the gutter)
As kids we learned Dear Liza too. Never hear it any other way. I would assume it's one of those regional things as the the specific name. A lot of children's songs are slightly different region by region.