Taxis, Cabs and Hackneys are all terms describing vehicles which are legally allowed to ply for hire - it is an offence for other types of vehicle to display the words 'Taxi', 'Cab' or 'Hire'.
Taxi operations outside London are controlled by local authorities under the provisions of various Acts of Parliament: the Town Police Clauses Act 1847, the Local Government (Miscellaneous Provisions) Act 1976 [which extends the 1847 Act], the Transport Act 1985 and bye-laws subject to Home Office approval.
If the taxi is waiting on a rank or at a stand, he or she must take the fare (unless there is a reasonable reason not to - whatever reasonable may mean).