At the risk of further thread drift here I note that the New Zealand Civil Aviation Act (1990, and amendments) is a lot clearer on the matter of a register:
S2 Interpretation:
New Zealand Register of Aircraft means the register of that name that is established under section 73
New Zealand registered aircraft means any aircraft that is for the time being registered by the Director under section 6(1)(a)
I won't go on to replicate the various sections, I doubt it would add much to this discussion, but it's good to see some jurisdictions attempt to provide a reasonably comprehensive set of definitions...