Watching too many TV crime shows will blind you to a stark reality: Witness testimony sucks. Look at the linked comparison of witness testimony in the Michael Brown shooting grand jury. Take any column, like the last one with number of shots fired. Its a total mess!
When videos emerge of police brutality, police defenders often say that video can lie. But I would argue that it is a hell of a lot better witness than the average person. My guess is that police like this kind of variation in witness testimony, because they know that in most, perhaps all, cases, they will be given the benefit of the doubt when the testimony conflicts.