I think they should say the following:
Day/VFR SEP flights.
[snip]
Will the CAA ever come out and lay the above down (or any variation/deviation that they might think fit?) No, of course not! They will claim that it's outside their terms of reference. Personally, I disagree, I think it's inside and they ought to do something.
TOO, isn't the situation you describe *exactly* the same as for any other private flight? You are PIC thus log P1, or you take an instructor and log it as Pu/t and the instructor P1. Since the CAA doesn't recognise the "club checkout" concept in any official document, but just consider it a private flight, why would they need to provide clarification as to the rules?