I’m an LE sharpshooter in a county with a lot a rural land. There is some discussion about the term “sniper” being used for law enforcement officers that use center-fire rifles with scopes for the purpose of shooting at moderate-to-long distances. Personally, I don’t get too worked up about being called a sniper, or a sharpshooter, or a marksman. I’ve never served in the military, but as others have alluded to, military sniper training is far more advanced than law enforcement training. So in that sense, "we are not worthy," but as 44AMP stated, "At one time, anyone who fired from concealment was considered a 'sniper'." It has happened as recently as the "D.C./Metro Snipers" a few years ago. So it's hard to fault people for referring to LE sharpshooters as snipers. Many citizens, the media, and even other law enforcement officers will refer to us “snipers.” It’s like people referring to “magazines” as “clips,” sometimes it's not worth your effort to bother correcting them.
I would agree with what Ben Towe had to say above, although, I would add that LE applications can happen in rural areas too. A good example would be that former LAPD officer that was barricaded in that cabin last year. So the ability to approach a backwoods hideout unseen, and wait for hours upon hours reconnoitering is important. As law enforcement our goal would be to apprehend a criminal so they can stand trial, but if a person poses an imminent threat of serious bodily injury or death to us or others then we will “end the threat.” This is all kind of common sense, but we would prefer to have absolute concealment, and good cover; however, because we are reactionary, getting both or either of these in a timely manner can be difficult. The average LE sharpshooter engagement distance is under 100 yards (some studies put the average distance around 55 yards), and almost all are under 150 yards. With such short distances we realistically wouldn’t have to worry about making adjustments to our optics, since point of aim and point of impact would only be off a couple inches up or down at most. With that said, we train for shorter and longer distances, and to make adjustments.
Getting more to the OP’s mindset question... You can probably guess or research the things law enforcement sharpshooters would have to train and be ready for, but you may not have given any thought about what happens after the shot. This really goes for any officer that kills someone in the line of duty. After a LEO takes the shot that’s generally the end of the ball game, as far as action goes. Then the LEO will have to worry about things like debriefing interviews, reports, psychological issues, and (the dreaded) grand jury. Assuming the grand jury decides that you shouldn’t be charged with murder, you have to continue living and working in a community where you killed someone, who most likely had family and friends in that community.