You raise some interesting issues. i can comment from my experience, which is fairly old. I expect there will be those with more current info.

I think a lot depends on your region. In southern Arizona, people hiking in the mountains were not typically lost. They knew where they were. The problem was to find a feasible route to where they wanted to be. If they did not have a map they were in deep doo-doo. Terrain is quite different in Wisconsin and other areas.

Pre cell phone, our operations were started by a phone call from someone who realized that Johnny was overdue and had been hiking in the mountains. i understand that nowadays many calls come directly from injured parties and they are able to give their location fairly well. This is good for operations, because you know the situation even before you are in the field.

I believe the Coast Guard prefers locations given in lat lon, but that may be simply because they are a maritime operation. Coordinates in nay system can be readily converted to another.

From experience, be active if a chopper or aircraft is involved. Signal with the best means available - a signal mirror is wonderful and a big bonfire works as well, although a trifle messy.

In mountains, at least, a compass is rarely useful, although I always carry one.

Hope we get some feedback from current SAR folks; practices are always evolving