The terminology for brushless, permanent magnet motors is confusing.
If you look in academic/technical literature like IEEE papers, then generally BLDC refers to brushless PM motors that have a trapezoidal back-emf and is driven by a six-step, trapezoidal drive, while PMSM refers to brushless PM motors that have a sinusoidal back-emf and are driven by sinusoidal waveforms. Be aware that brushless motors with a trapezoidal back-emf can be driven by sinusoidal waveforms and vice versa. And also be aware that trapezoidal and sinusoidal back-emf's are ideals and you can never really get either one. Of course, I've also seen IEEE papers that refer to BLAC motors and use other terminology, so this isn't strict across the board.
Industry hasn't really adopted this terminology completely. You often will see companies refer to BLDC motors, as you've already pointed out. And generally by BLDC they mean exactly what the academics mean - a brushless motor with a trapezoidal back-emf. However, I've also seen these referred to DC brushless (DCB) motors, brushless PM (BPM) motors, or even PMSM's.
With what academic literature refers to as PMSM's, I've seen them called PMSM's, brushless AC (BLAC) motors, AC servo motors, brushless servomotor (BLSM) and others.
Some manufacturers may not make a distinction between the 2 because in reality it isn't an either/or thing. You can't make a brushless motor with a perfect trapezoidal back-emf and you can't make one with a perfect sinusoidal back-emf. Your best bet is to talk directly to manufacturers and tell them what you want to do and they will guide you in the right direction.
In reality: Most so called BLDC motors on the market have sinusoidal
back EMF, and can be controlled by the same FOC method as PMSM motor.
But I think they are still BLDC motor, not PMSM.
This may or may not be true. In my experience, BLDC motors do not have sinusoidal back-emf; they are much closer to trapezoidal. Keep in mind that we are talking about the phase back-emf, not the line-to-line back-emf. Sometimes the line-to-line back-emf looks close to sinusoidal while the phase back-emf doesn't.
Don't let inductive current lag (apparent power) confuse the issue. Ohm's law still applies.
When you apply a voltage to a motor, the resulting motion generates a "back ElectroMotive Force" against the coils; its rotation feeds back a resistance that regulates the current, in proportion to the motor speed.
If an external load reduces the speed of the motor, that "back emf" is reduced (the resistance of the motor decreases), increasing current in order to maintain the voltage, and thus power increases.
Best Answer
This increase in current when you load a motor more is not related to inductance. It is mostly because of the mechanism of conversion of electric energy to mechanical energy, which follows Lenz's laws (among others). There is no change in inductance happening.
When you increase the load on a motor, you are in effect putting more load torque on the motor, which opposes the torque produced by the motor. This motor-produced torque is directly proportional to the current flowing in it. You applying the load reduces the speed and draws more current. It is like you lifting a 5 kg load up a hill and then someone puts another 5 kg weight on you. You would need to supply more of your muscular energy to maintain the same speed, just like the motor draws more current.
The inductance present in the coils is due to the leakage inductance, which itself is present due to the imperfect (rather incomplete) coupling of the magnetic fields of the rotor magnets and the stator coils. This remains constant and is largely fixed by motor construction.