A local council is a corporate body and by and large the individuals involved with it have no power to act independently of the legislation that defines local council activities. For the basic rules of procedure, that means the Local Government Act 1972.
The case of the clerk is quite complex, because the clerk is normally appointed as the proper officer for just about everything, and the legislation refers to the proper officer. The clerk is usually also the responsible financial officer (RFO).
The only ways for council decisions to be properly made is one of three possibilities:
- A properly constituted council meeting where 3 clear days notice have been given and an agenda published showing the public what matters may be decided on
- A properly constituted committee or subcommittee meeting, notice and agenda as above, on matters delegated by the full council
- Decisions by the clerk based on delegated powers determined by the full council. There are some cases where the clerk should act anyway in the interests of the council, and there are some common law precedents that are not visible in the statutes.
None of this really supports the idea that there can be an "executive". There is a grey area, in that it is reasonable for the clerk to discuss matters needing a decision with councillors, and the clerk might choose to give particular weight to the views of the elected chairman and vice chairman. However, the decision must in the end be that of the clerk, and usually must conform to the delegated powers granted by the council.
Personally, I don't recognise the existence of an unwritten understanding on executive positions, and don't believe that decisions could be made on the basis of an unwritten understanding.