I think they thought they were being cute with Archie - a non-traditional, historically working class name or a name from comics - that fitted in with their narrative of being modernisers, not following protocol and not bothered about titles - the "just call me Harry" phase. Perhaps they were hoping for some sort of backlash or drama from the RF that didn't come.
By the time Lili was born they were desperate for royal adjacency, were definite about using their Prince/Duke/Duchess titles, being announced by them, cosplaying pseudo royals with monograms, faux tours etc etc. So they used QEII's private name ...
This is a really interesting point. Lunde. I agree that the children's names reflect the HAM's instability - their constant 'change and change about' character.
Personally I'm not a fan of cutesy 'original' or pet names for children such as those chosen by Geldof and Yates, or Jonathan Ross and Jane Goodman. These children will grow up and become middle-aged and elderly people, they'll go through all kinds of experiences and they might want a serious career. Give them a less stand-out name and allow them to choose their own short or pet version.
It wouldn't surprise me if Lilibet decides to become Elizabeth, or something entirely different, when she's older. Diana would have been a better first name (I know it's her middle name) in preference. I think Archie's too casual and trendy for a prince, but as you say, they were in that particular phase when they named him.