UK police departments are not funded by councils. They are funded in two ways: A precept issued to council tax collection authorities, and grants from the Home Office.
Note that the precept is set by the police, not by the council - the councils have no authority over police spending.
>UK police departments are not funded by councils. //
When council tax has an earmarked additional portion specifically noted to be for police funding I think you're clutching at semantics. UK Police are in part funded from council tax garnishing, a precept as you say.
Where I am at least [in the UK] the police commissioner details a request based on a proposed budget which is put to a panel "the police and crime panel" which sit to set the funding that will be made. The panel comprises members of the local councils and they must vote as to whether to endorse an agreed budget and so collect the tax to pay for it.
The distinction you seem to be making doesn't appear to be there. The police and council set the budget between them effectively, councillors having veto powers, the council collect the money as part of council tax.
Note that the precept is set by the police, not by the council - the councils have no authority over police spending.