You definitely want a headlamp as your primary light. While it is easy to hold a headlamp in hand when desirable, it doesn't work so well the other way around.

Battery compatibility is highly desirable, not only within your light array, but also within the team.

If I were going out the door right now, I would definitely carry my Zebralight 502 - light and bright. I would augment with a thrower - powered by AA batteries, in my case. In reserve I would have my EDC, small and handy, and just right for changing batteries, etc. (probably a Gerber Ultra Infinity)

Your light array will be an essential part of your gear. There seems to be some regulation or other that requires all SAR ops to occur in the dark. There weren't very many of the 450 or so in my experience that occurred exclusively in the daylight.

Considering how small LED light are these days, there is no good reason you could not carry more than three, but definitely carry the types I have indicated. You will use them a lot.