From reading the statutes and personal observation it does not appear to matter whether a vehicle is owned by a department or is privately owned -- it can be an "authorized emergency vehicle." For example, a volunteer firefighter's vehicle is an authorized emergency vehicle if it is equipped according to the law and is on its way to a fire alarm or medical emergency. (And failing to yield to it brings the same penalties as failing to yield to an ambulance or police car. That is, if there is ever a cop around to witness it.
) I know some LEOs who have privately owned vehicles set up as police vehicles; one a new Dodge police model, the other is a pickup truck. Interestingly the Dodge is "unmarked" although obviously a police vehicle (police bumper/grill guard, plain white, blackwalls) with lights mounted in the interior, but the pickup is clearly marked with "POLICE" decals. The owners work for small city/town police forces and drive city vehicles on duty, but they also work security and traffic control (e.g. construction sites) on their own time, and use their own vehicles to do so. I also believe that at least some of our local constables buy and outfit their own vehicles.
There are specific requirements in the law for equipping an authorized emergency vehicles, but there are several exceptions and variations for police vehicles. Also there are rules for when the lights and siren should be used, but again, law enforcement exceptions do exist.
I don't think there are any requirements that a police vehicle be new, or not "beat up;" some of the local county sheriff vehicles are not what one would call "pristine."
So the point of all this is that the condition or the position or type of lights on the vehicle in the OP really don't have any bearing on whether or not it was really a police car, which is unfortunate in that it can make it difficult for a citizen to be sure. The yelling and gesturing business was certainly unprofessional and deserves to be reported -- is you can ever figure out whose cop it was. Too bad you didn't have a passenger with a video camera or phone -- would have made a great Youtube flick.