I hope that "Family" never appears on any street signs, but it might be legitimate on a map/leaflet highlighting things like the NEPN, Innocent etc.
The crucial word is Network.
I hope the idea of an (the) Edinburgh Cycle Network does get some prominence, perhaps with Sustrans style signs and also road markings - basically what was happening in London prior to the "Super Highway' concept.
Going back to ATAP - "it will need to cross the city centre east to west and north to south".
Perhaps these should be quite clearly created/marked/marketed - Cross Town Connectors.
These will certainly be much more than 'family friendly' routes. I would hope that they will be designed/created well enough to be attractive to people who happen to want to cycle in the city with youngish children.
I seem to remember general aspirations (not necessarily in ATAP) for facilities that could be used by 10 year olds - on their own.
To repeat - the crucial word is Network. So anything that is designated Network must be conveniently and safely linked up. This is currently the case for NEPN, because it's mostly off-road. It may soon be possible to refer to routes between and Portobello and Leith as the East Edinburgh Cycle Network. But neither could really be considered to be part of an Edinburgh Cycle Network unless they were well connected to each other - and they would really need to be linked to paths and decent on-road sections in other parts of the city.