Spur. A branch from a ring or radial final circuit.
Circuit. An assembly of electrical equipment supplied from the same origin and protected against overcurrent by the same protective device(s).
Those are, indeed, the BS7671 definitions, but IMO they are totally unsatisfactory. In particular, to say that a branch from a radial (with same cable CSA as the rest of the circuit) is a 'spur' is totally meaningless - if two 'branches' arise from one point in a radial (say, a socket), which is a continuation of the radial and which is the spur? That's daft!
OK so how is a cable coming from a FCU called a spur? It forms it's own circuit as it has it's own over current device but yet we still call it a spur.
Indeed. What you say is true in terms of the BS7671 definitions (and, as you know, creates a potential issue in relation to the new notification requirement {for a 'new circuit'} in England). If the FCU is connected to a ring circuit by a bit of single cable (rather than being wired-in as part of the ring), then the cable from ring to FCU would obviously qualify as 'a spur', both per BS7671 and 'common sense' definitions. To call the wiring downstream of the FCU as 'a spur' is, I suppose, just ('convenient') 'established jargon'.
The situation is obviously more straightforward with a ring final than a radial final circuit. It seems very reasonable to regard any single-cable emanation from the ring as being 'a spur' - and (despite the BS7671 definition of 'a circuit') that sensibly also apply downstream of any FCU in that 'single cable' circuit.
With radials, it's different. As above, if all cable is the same CSA, then any attempt to define a 'spur' is totally meaningless, and unhelpful. Personally, if there is a reduced-CSA branch of a radial (fed via an FCU if it feeds more than one socket), I would be inclined to call that a spur. That might actually suggest a more workable/useful definition of a spur, from either ring or radial circuit - one could define it as a single-cable reduced-CSA branch from a ring or radial final.
Kind Regards, John