It is the firm rather than the Electrician who is registered as an NICEIC Approved Contractor. However the firm will have (a) Qualifying Supervisor(s) (who most hold relevant technical qualifications) and of course a Principal Duty Holder (who is the boss). NICEIC Approved Contractors are required to have sufficient insurance; copies of relevant Standards; Health & Safety Policy; Risk Assessments etc. and are audited on all of this, and also inspected on their work and assessed.If I'm wrong someone will correct me, but AFAIK the former, "Approved Contractor", is much less trivial than DI, and does require "proper" City & Guilds qualifications.
It is - as you rightly suggest - not trivial.