No. The section is referring to proceedings being instituted in the County Court if the court has a land jurisdiction or in the High Court in any other case. It can only mean to refer to the civil claim for harassment etc It's talking about which of the two civil courts the proceedings are started in. Otherwise we wouldn't know whether they had to be started in the High Court only, whether a Registrar can deal with them, and so on.It's impossible to start any criminal proceedings in either court. Criminal proceedings start in the magistrates court and either stay there (summary) or are sent to Crown Court (on indictment).
The case in the Supreme Court was on a different point than that which you would make. It was whether the FSA could not institute a prosecution for an offence when it expressly could for a very similar one under the same act. Held: It was plain that Parliament intended the FSA to institute prosecutions for both.
Part III does not say 'may only' be prosecuted, true.But it appears that it is defining which of various possible local government or other bodies may prosecute, not giving some extra power to District Councils which they wouldn't otherwise have when private individuals do have it [see judgment in the case you cited for general powers to prosecute ].It therefore seems an exclusive power and the absence of the word 'only' ,though it might have made that plainer, does not affect that.