Hello Rob
The only reference I have to New Zealand buttons is Corbett's book "The Regimental Badges of New Zealand" which includes 5 pages of buttons.
Much of what is illustrated agrees with what Tinto has already said - 'standard' British single gun Royal Artillery button in white metal for "Volunteer Artillery 1870"; "Dunedin Artillery 1870" (interestingly the gun on this button points to the viewer's right!); "Territorial Artillery 1911" and "NZ Artillery 1911" (your third button).
Corbett does not illustrate or mention the NZ marked British style Artillery button.
He also illustrates a button for "Naval Artillery 1903" but the photo is not very clear although it appears to be the Gothic letters 'NAV' spread across the anchor of a normal naval button of the period.
Roger
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