The system of classifying social groups, which you quote, is no longer used by government statisticians. They now use a 9 point scale, as follows:
1 Managers and senior officials
2 Professional occupations
3 Associate professional and technical occupations
4 Administrative and secretarial occupations
5 Skilled trades occupations
6 Personal service occupations
7 Sales and customer service occupations
8 Process, plant and machine operatives
9 Elementary occupations
Under this system, a chef is coded '5434' where the first digit indicates which of the above groups applies. So, a chef is defined as being in a 'skilled trades occupation'. Under the older scheme, that would seem to suggest either group III(a) or III(b). Since the occupation is largely a manual one, that would seem to indicate that group III(b) is the appropriate one.
Note that I've used (a) and (b) to indicate the subdivisions of group III. Statisticians normally refer to groups III(wc) and III(man), representing 'white collar' and 'manual' respectively.
Chris