These days, police departments have pages and pages of regulations governing the circumstances under which deadly force may be employed. I suspect that lawmen in the Territories were not so encumbered. But that does not mean there were no such rules, written or not, then in force either by dictate or custom.

As a hypothetical, suppose you are a USDM hired to guard a train. At one of the stops you leave the car and walk down the platform. You notice, only yards away, that a robber is holding the conductor at gunpoint. The robber doesn't see you and you have a clear field of fire. Do you take the shot? Are you obligated to call out to the bandit, and give him a chance to surrender?

--meursault

In your example, since it is in a public place, I'd venture most officers would identify themselves and call for a halt to the action. 'course the call and shot might be pretty close together. I've noticed that most instances of the use of violent force were downright ambushes in the boonies and more proper in public view.
I don't know if the Marshals had a formal code of conduct, but I suspect they did. Be interesting to see it.
The Indian Police had general rules and regulations. Every member must...be orderly and respectful in deportment. As to an arrest,or interfering with a citizen--the constable, before making an arrest, had to ascertain that the offense charged constitutes a crime or misdemeanor for which a person can be lawfully detained, and the ground of the charge is reasonable. The rule dealing with force was: No member shall maltreat or use unnecessary violence toward a prisoner or other persons; any member may be dismissed for such.
Your example, at a distance, clearly fits the guidelines for effecing an arrest but the use of violence has yet to prove itself necessary, so again a warning is necessary.
Old Judge Parker, and even the court at Paris, Texas, really got after the deputies for shooting "prisoners trying to escape," especially when all the deceased citizen had was the deputy's assertion he was the law and that the citizen was a prisoner because a stranger said so.

This is an interesting question, "are you obligated to call out to the bandit." The only thing clear is that the deputy is obligated to protect life and enforce the law. His obligation is to protect the life of the conductor, who has a gun pointed at him. To me, the issue for the deputy is not about shooting the robber in the back, or whatever, or giving a verbal warning. The issue is trying to prevent the conductor being shot and killed. The deputy is forced to judge how much force to use.