Short answer:

Makes perfect sense in my opinion, this is how I would expect it to work and I bet everyone already has it configured this way.
I would go with mode b).

Long answer:

There are two separate cases in mode b):

- If you don't have the background daemon running, you are probably starting the main program in order to update your position and to see the map/check in - daemon should be on while the main program is running.

- If you already have the daemon running, you are probably starting the main program to see the map and/or check in - in this case the daemon should either:
b1) keep the same settings it had in background mode, or
b2) restart with the default settings

My humble (and possibly too laborious) suggestion is:
Keep the background daemon and the main program daemon separate from each other. When the user starts the main program, you could suspend the background daemon until the main program is closed. The settings should be separate as well, as the use cases for using the background daemon and running the main program are typically rather different.