Nah, nah ... I don't think the "James Barrowman" one is yours.

[I'm sure your one is just "plain" James HENDERSON - no middle name).
Had just posted the info re: Thomas Herbert / Thompson Herbert" to show that they were two different people and that Thompson Herbert HENDERSON although buried in the area where your James lived, was less likely to be a son of "a John and Jane HENDERSON":
But don't splash out on buying any printouts (certs.) just yet.

I'll check the death microfiche also (for place of registration) - have narrowed deaths down to a handful of likelies.
Found this on paperspast
Otago Witness , Issue 458, 8 September 1860, Page 4 Article view
Shipping News.
Early in June [], the Henrietta and Robert Henderson, [was] despatched from the Clyde by the agents of the Provincial Government of Otago, with full complements of passengers. Besides those assisted by the agents, a large number not only paid their own passages, but took out with them considerable sums of money.
ARRIVALS.
3 Sept 1860 Arrived in Dunedin from Glasgow aboard the Henrietta & Robert Henderson
..............John Barrowman and wife........... William Henderson, Agnes Henderson,..............John Henderson and wife
DEATHS [on Board]........ June 26, Ann Henderson aged 20 years, of scarlatina...........
this then could be maybe a cousin or brothers that came out together,do you think?
karen