Sinusoidal and trapezoidal back-emf's on brushless motors are idealizations and on real motors you will never find either one. Generally, when a motor with a trapezoidal-like back-emf is driven with rectangular pulse currents, it is referred to as a BLDC. Also, generally, when a motor with a sinusoidal-like back-emf is driven with sinusoidal currents, it is referred to as a BLAC. However, you can drive motors with trapezoidal-like back-emf's with sinusoidal currents and motors with sinusoidal-like back-emf's with rectangular currents. I see the former (sinusoidal back-emf with rectangular currents) quite often. Duane Hanselman's book Brushless Motors: Magnetic Design, Performance, and Control has a good discussion of all this and the back of his book shows what the line-to-line and phase back-emf's should look like for various motor topologies. Hendershot's Design of Brushless Permanent-Magnet Machines also has a useful discussion of this topic.
With those caveats in mind, in order to identify the back-emf shape you will need to back-drive your motor with another motor and then observe the back-emf waveform by measuring it with an oscilloscope across two of the leads. The shape you see is your back-emf.
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.
Best Answer
A perminant magnet synchronous machine (PMSM) must be excited with an AC current for torque to be developed.
A Brush less DC (BLDC) PMSM has a backEMF profile which is trapizoidal in shape to maximise the peak torque but also to simplify the drive algorithm as they can be excited with a simple quasi square wave controller.
A Brush less AC (BLAC) machine has a sinusoidal backEMF profile and thus should be excited with a sinus current waveform
There are three types of BLAC machines
Surface Permanent Magnet (SPM). The rotor magnets are bonded to the rotor
Interior Permanent Magnet synchronous machines (IPMSM) where the magnets are mounted in slots of the rotor
Flux switching Permanent Magnet (FSPM). A cross between an SR and a IPMSM.
Fundamentally they are controlled exactly the same way. However due to the fact there is a reluctance component associated with the IPMSM, the machines requires a slight amount of phase advance to provide a balanced torque profile. The motor also benefits from reluctance torque and thus can be more efficient than a SPM as it can produce slightly higher torque for the same current. Likewise they can operate at higher velocities due to the magnets are encased by the rotor