Yep, Permanent Magnet AC (PMAC) motors are common types of synchronous motors. They are generally more compact than induction motors for the same power output. Induction motors are generally less expensive.
DC Brushless and PMAC are interchangeable. Oh, and nowadays, you don't need sensors (hall, encoder etc). 'Sensorless' motor control is possible by sensing the back emf of the motor or reconstructing the motor current via (super inexpensive) shunt resistors on the legs of the inverter.
I actually wasn't aware that induction motors don't use permanent magnets. RC model aircraft is a hobby of mine, and all I knew is that the commonly-used brushless AC motors certainly use permanent magnets.
As a first order simplification, motors are all about pairs of magnetic fields attracting / repelling each other. Often, one side of the pair is a permanent magnet. If instead the magnetism is induced (implying no magnet involved, as the other side must be switchable / controllable / varying, thus not a permanent magnet), you have an induction motor.
At low power levels, permanent magnet motors (specifically brushless motors) tend to have higher efficiency than induction motors, additionally, they don't require complicated control systems to operate efficiently at varying speeds. Thus you are unlikely to see an induction motor in an RC model anytime soon.
Stepper motors simply provide accurate and fast occasional rotational positioning. This requires high torque at a wide rpm range, including 0 rpm, for obvious reasons. stepper motors are a very bad application for induction motors, since everything other than steady state operation requires advanced control systems and switching power supply circuitry. For the basic induction motor without an advanced control system, the starting torque must be provided by inefficient means, and is still very limited.
Synchronous motors are motors where the rotation of the magnetic field is synchronous with the rotation of the motor. For all induction motors, the induced field is a result of a varying magnetic field across the element in question. This implies that the speed of the rotor must be different than the speed of the magnetic field. Thus, no induction motor is a synchronous motor, almost by definition.
