yep, thats what i meant, good possibility that Walter is Abes son.
and that William R on the 1881c isn't William Robert, tho he was born 1865 in NZ.
Interesting morning,
No William Robert Martin that fits the bill.
No William Robert Cook(e) 1859 - 1869
but
Now fairly sure the change to COOKE was a turn of century thing.
Possible births:
1896: John Eric Norman COOK [262] New Plymouth
1897: William Robert COOK [...] New Plymouth
1901: Aner Charles COOKE "
Possible marriages:
1921: John Eric COOKE m Margiritte Louise MOONEY [06402]
1924: William Robert COOKE m Annie Elizabeth NORTON [09972]
E. roll 1924
John Eric COOKE Clerk Bretts Ave, Takapuna
Madge COOKE married "
Now fairly sure William was not a birth child of Aner and Charlotte.
I believe the MARTIN addition
after his parents deaths is definitely some clue and if John Eric Norman is Williams kid, I think another clue.
Maybe you should try for J E N Cook birth or marriage ?
my 2 cents and your 15 minutes,

Good Luck