Firstly it is not the coppers interpretation that decides what is a weapon, it merely decides what they suspect to be a weapon and it has to be in the public interest to pursue a prosecution. As I am sure the black belt barrister will remind you on you tube, in order to secure a conviction it has to beyond reasonable doubt that a crime has been committed. It would be highly unreasonable to dismiss the grounds for carrying a knife or axe if there plentiful evidence of its intended lawful use. I think the odds on most housing estates are that you are more likely to be threatened with an unlawful knife in the absence of any visible police than you are to be stopped and searched when you have a clear reason for posessing tools lawfully. I have before retirement been in positions which require an enhanced criminal DBS disclosure and have never worried about it.
police do have powers to deal with incidents that fall short of a full prosecution and involvement of the CPS. They may confiscate items or issue a police caution. But I get your point