It's not that simple, as any legal system will show. There must be good reason for me to believe that the person (in this case a cop) has an intention to kill me. Normally I would not make this assumption just because a policeman pointed a firearm at me. On the other hand, if the cop had just shot at me unprovoked, I might consider shooting him or her. Of course ideally in that situation one should aim to remove them as a threat without killing them.

Here is a starting point for this definition:

Violence is the use of force against another person, when the force used is more than necessary for preventing a greater act of violence.

I haven't read all the arguments, and this is my first posting here. So excuse any misunderstandings.

I am inclined to agree with your train of thought. My preliminary formulation would be something like:

"Violence is the use of physical force against another person, but only when the force used is more than that necessary to avoid a greater act of violence"

This formulation is perhaps similar to the way regulations of police use of force are formulated, they are limited to the use of reasonable and proportionate force. They would argue that by following this guideline they are not being violent even in quite aggressive and heated confrontations.

I support the statement "The doing of justice never requires violence" in the sense that I am inclined to consider any use of violence as inherently unjust. However I think that there are times when, for example, widespread use of force in overturning a despotic regime can be justified as preventing greater acts of violence against a population.