As long as the content of the Cert is to BS7671 it doesn't matter.
I cant think of anything specific with an NICEIC cert that would make me think it was more detailed? Maybe r1, r2 & rN figures, along with supply figures when not directly connected to the origin.
I myself find it all necessary to record. For my reports that I designed myself I also include R1&RN, as polarity cannot be confirmed without it, and some electricians believe it is not a necessary test as it isn't on a report.
As for smaller completion certs, no. My own can be adjusted, but third party certs will not be likely to manufacture single circuit installation certs. Some software may accommodate it, but I wouldn't know.