Laws vary by state, and just as I always do when I reply to a thread as an LEO, a reply as a South Carolina LEO.
The act of impersonating a police officer is not dependent on wearing a badge that says "police", or stating "police" during an incident. If a person intentionally acts in a manner that would lead a reasonable and prudent person to believe the the person is a law enforcement officer, then that person could be charged as such, based on the totality of the circumstances.
When something goes down nobody sees what those 1/4" tall letters on a badge say.
Case in point:
We had a volunteer fire fighter from a local fire department, which issued badges. He was apparently behind a car which he felt was driving recklessly. He turned on his red lights, waited until the driver pulled to the side to let him pass, and pulled in behind him, got out, approached the vehicle, held up his fire dept badge, then began berating the guy for his driving. He was charged with, and convicted of impersonating, even though he never said he was the police, a held up a badge that said the name of his Fire Dept, and "firefighter".
Anyone can buy one of those badges, if they were only available to people that actually have a ccw then it might be different, but the various CCW badges, and security badges, get taken off drug dealers and repeat felons on a semi-regular basis. It's not a epidemic problem, but it happens with enough regularity that most cops should be aware of it.
Cops are used to, and train for, coming across ccw holders, armed home-owners, off duty and/or plain clothes officers from their jurisdiction, an overlapping jurisdiction, and even jurisdictions a 1000 miles away on vacation (that one's kinda rare), and people pretending to be all of the above.
Next point is that badges saturate our society. Beyond LEO's, you have firefighters, EMS personnel, security guards, loss-prevention personnel, repo guys, bail bondsmen, code enforcement, tax inspectors, building inspectors, bailiffs, process servers, IT security geeks, alarm system techs, private investigators, the list goes on and on, who carry badges in some locations. This means that badges don't carry the weight they used to.
All law officers are trained that the ones who show up in marked cars and wearing uniforms are in charge until the situation is under control. Your actions are what the response is going to be based on when law enforcement arrives on-scene, people that appear to be a threat will be dealt with as such, those that immediately yield and follow directions will still be around to explain themselves, regardless of a chunk of metal on their belt or in their wallet.