My question is: Should we consider each of these as a different name of Allah? Or just extensions of the name Ar-Rabb?
It is one name, (Rab), but it is mostly mentioned added to his creatures:
Ar-Rabb >> The Rab
Rabbi n-nas >> The Rab of people (Lord of people)
Rabbil Aalamin >> The Rab of the world (lord of world)
Rab Al-Malaekah >> The Rab of Angels (Lord of angels)
They all has the same indication that he is the Lord and owner of all his creatures and the one who can manipulate and control them.