(There are some reports both animals were mules.)
"Henry led in the work," said museum spokeswoman Nicola Bennett. "He was there on day one, he came up with the idea, he was instrumental in setting up such work.
"Our guy Henry made it right through the Gallipoli campaign and returned home ... We don't want to start any controversy about Simpson's efforts, but it's an opportunity to recognise our Kiwi - doing something before Simpson."
Both men landed at Anzac Cove on April 25, 1915. Henry won his medal for his actions on that day.
Simpson's exploits were recorded starting the day after, Ms Bennett said.
The painting in question, by New Zealander Horace Moore-Jones, was done from a photograph after the artist returned to Dunedin.
The man in the photograph, to add to the confusion, was New Zealand medic Richard Henderson, who took over Simpson's work after he was fatally shot.
There are several copies of the painting and they are referred to by various titles: "The man with the donkey", "Murphy and his donkey", or "Simpson and his donkey".
Ms Bennett said the museum had uncovered reports that Henry had earlier posed for Moore-Jones and he was the real "Murphy".
Henry has previously been mentioned as an early adopter of donkeys as ambulances, and old clippings upon his death in 1950 suggested a link between him and the painting. But the story never gained traction and was rediscovered by the museum only when it was preparing an exhibit.
John Simpson Kirkpatrick was born in Britain but later moved to Australia.
In August 1914, he enlisted in the Australian Imperial Force and served at Gallipoli the following year.
He was killed by machinegun fire while carrying two wounded men and was buried on the beach at Hell Spit.