Yes that is correct. Your GNSS receiver can use the GPS (formerly NAVSTAR), GLONASS, Galileo, Beidou, or any combination of those systems satellites depending on the receiver's capabilities to calculate its position and state vectors, even without SBAS. SBAS without the GNSS satellites however gives you no position. SBAS typically improves the position accuracy by an order of magnitude or better (e.g. 10m precision to <1m precision), and provides additional system integrity information.
The predominant error source for position information with GNSS is in the signal path delays due to the variability of water vapour in the atmosphere. Water vapour affects the refractive index of the atmosphere at radio wavelengths, hence causes the signals to not travel in a straight line, thus the signal travels a longer distance than the assumed straight line. This translates into what's called a "path delay". The ground monitoring stations for the SBAS know their exact locations, and can reverse engineer the approximate path delays over a larger area (not just where the reference ground station is located), which in turn can be used to model the water vapour distribution in the atmosphere and depending on the receivers position, a correction factor for each signal path can then be calculated. This is also useful for other things like radio astronomy.