I control the MR system and all systems adjacent to it. I "control" a system
if I have at least one (non-fighter) ship there and I control every planet
in the system = 3 vp

For the above objective do "empty" systems( withour planets) still need to be occupied?

Yupperdoodles.

For this objective, you need to control MR (the planet and the space) as well as all adjacent systems. System control requires a non-fighter ship in space and control of all the planets. Systems without planets only need the non-fighter ship in space.

Special systems that are impossible to occupy, such as Super Novae or Asteroid Fields, would be excluded from the requirements of the objective.

Destroyer spam, go go go.

Short answer to the OP's topic question: A system = a hexagonal map piece.

I take it Saar do not need to control adjacent Asteroid Fields once they managed to find out how to stop in them

(There's noone else going to be inside ever anyway - so technically they are passively in control of all Asteroids anyway.)