I couldn't find any single resource that would answer all parts of your question. Here's what I could piece together from multiple sources.
A register-based census is a type of census that relies on already existing population registers (according to Google's dictionary function, a register is simply "an official list or record, for example of births, marriages, and deaths [...]"). See e.g. this description in Wikipedia of the new Swiss census format (with which I have some first-hand experience): "In order to ease the burden on the population, the information is primarily drawn from population registers and supplemented by sample surveys. Only a small proportion of the population (about 5%) is surveyed in writing or by telephone." The way this works in Switzerland is that the municipalities and the states (cantons) already have very detailed population registers since the population has to report all changes in their residence and in their civil status to the municipalities concerned. The Federal Statistical Office can thus rely to a large extent on this information.
The register-based census seems to stand in contrast to a traditional census, in which the entire population has to fill out and return a census form. An example of such a traditional census is the United States Census.
This distinction fits what Statistics New Zealand describes as pre-conditions for a register-based census in the document you linked to: " a strong legal basis, public approval, unified identification systems, and comprehensive and reliable register systems developed for administrative needs."
- strong legal basis and public approval: the government would need to be allowed to use this data, which has hitherto been collected for different purposes that may not include censal purposes
- unified identification systems: if Statistics New Zealand is going to pull together population registers from different sources, it will have to be possible to somehow match them (e.g. what if one dataset includes middle names and the other one doesn't?)
- comprehensive and reliable register systems: this one is obvious; a switch to a register-based system should not impact the quality of the census (in the Swiss example that I mentioned above, municipalities have long since maintained accurate population registers, but, for various reasons, that is not the case everywhere)
After reading the description in Wikipedia of the New Zealand census, it seems natural that they would at least consider switching to a register-based census: "all census forms are hand-delivered by census workers"
A statistical spine is a concept related to the register-based census. When you are pulling together information from various data sources, data points will need to be linked and checked for duplicates and omissions. E.g. the health data on John Smith should ideally be linked to the same person's tax data, etc. The spine thus describes this unification of different registers. The analogy to the biological spine probably comes from the fact that you have one variable that all or most datasets have in common, such as people's names, "in the middle" and there are then various other data sources attached to each name. See e.g. this document by the UK's Office for National Statistics (search for "spine" and start reading at the first appearance) or this sentence from the New Zealand document that you provided: "Within the statistical agency, the population register serves as a central spine and establishes the reference population of people resident in the country."
As for the term index, I think it is probably interchangeably used with register and dataset but I'm happy to be proved wrong.