The first says you need to be told “climb now” the other says any climb clearance not insignificant.
The first is more restrictive, yes. Do the UK ATC give clearances other than “climb now” or “climb via SID”?
To answer your question though, my opinion is that a description of ICAO differences does not override the actual rule that is different. ICAO differences in your airway manual are there as a quick reference to the rules in a specific airspace, but they aren’t the rules and might miss some subtleties.