There are more books that offer "South Australia," however I'm not sure how original those are. Also, by way of disclaimer, I have not heard the Carpenter or Gordon recordings, so I don't know their texts and tunes.

However -- Looking at the evidence and turning attention to the question in the title of this thread:

LA Smith had:
"Heave away, you ruler king,"

Hatfield:
"Hooray You're a lanky!", which I've suggested is a mondegreen of "hooray you ruling/rolling king." Verses include, "What makes you call me a ruler and king?/‘Cause I'm married to an Indian queen,"

Parrish:
"Haul away, I’m a rollin’ king"

Doerflinger:
"Heave away, you ruler king,"

Harlow:
"Heave away, you Ruler King,"

Baldwyn:
has verse, "Have you seen my bowery queen?"

Colcord:
"Heave away, you rolling king,"

It certainly doesn't provide any clear answer. If I may hazard an interpretation: "Ruler/ruling" seems to dominate. The only ones with "Rolling" are Parrish and Colcord. Colcord's I am suspicious of; I need to know more about how she obtained it and what she *might* have inferred, since it is not a strictly field-collected version. That leaves Parrish's Georgia Islands singers. They are the only ones to make the phrase "I'm a..." rather than "you..." This somewhat limits the meaning of the phrase "rolling king." My instinct tells me that this version is more likely to have been a mondegreen.


Additional note:
Doerflinger's is the only source I've seen to use, the chorus with
"Heave away, don’t you hear me sing?" It is a distinct chorus form.

I know AL Lloyd's source is a mystery (?). However, if it was not based in an oral source, it looks most likely it was based (chorus and tune) off Doerflinger's book. It has the "hear me sing," and, if one does not read the very convoluted notation meticulously, one can get the tune he used.

BTW, Lloyd sang "lollop around Cape Horn." The Clancys in 1962 probably introduced "wallop".