I downvoted this, as this rule would force people do something that would eventually lead to the disencouragement of a lot of people, leading eventually to leave the server, disband of a faction and so on and so on. Not to mention that sometimes you don't want to be recognized when doing some activities.
You took police, NHS and park rangers as an example. All of them are allowed to go off duty and do whatever they want dressed up as they want, and I wouldn't see a reason why It would be bad to be able to do so.
Now the OP talks about pushing turfs, for me it means that once you are pushing a Turf, you should be close to if not already in a war. As I said previously, each official war has its terms that are agreed upon from all the involved parties and approved by staff, so the core of the server doesn't get disrupted.
If a player is involved in a situation like this, you should be able to talk with the faction leader, they are indeed there and have extra privileges, to ensure that every gang/faction member is educated if wrong doing is/was done.
Those not involved, should just get away from the situation and it could out your life at risk, which is a total different topic.
That said, I see no problem if a gang is pushing another out of colors, that would be their business.