Why is a bandpass filtered analog PM signal still PM and not AM I know, that in theory a sinusoidally phase modulated (PM) signal with an expression like $\Re(e^{\mathrm{j}k\sin(\omega_s t)} \cdot e^{\mathrm{j}\omega_ct})$ with the signal frequency $\omega_s$, the modulation index $k$ and the carrier frequency $\omega_c$ has frequency components at the frequencies $\omega_c \pm n\cdot\omega_s$ with amplitudes calculated from the bessel functions of the first kind.
This means, that for no noise and for an infinite timespan, the fourier transform of such a PM signal consists only of a carrier and (an infinite number of) symmetric spurs around the carrier.
Now when I filter this PM signal using a bandpass with center frequency $\omega_c$ and bandwidth slightly larger than $2\omega_s$, I get a signal with a carrier and a single pair of sidelobes/spurs that are symmetric around the carrier. When I try to demodulate this signal (either using a dsp simulation or using a signal generator and a signal analyzer), I get almost no amplitude modulation and a roughly sinusoidal phase modulation.
I am wondering about two things, I don't understand:

Why is the bandpass filtered signal still a PM signal and not an AM signal? AFAIK, a sinusoidally modulated AM signal consists of the (optional) carrier at $\omega_c$ and two sidelobes at $\omega_c\pm\omega_s$. But this seems to be exactly the spectrum of the filtered PM signal. What am I missing here?
How are AM sidebands/spurs different from PM sidebands/spurs? Given I have a pair of spurs around the carrier at $\omega_c$ with offset $\pm\omega_s$, how can I determine if those spurs are a sinusoidal amplitude modulation or a sinusoidal phase modulation?

Edit 1:
Based on Fat32's answer, I played a little bit with different phase angles between spurs. In particular, I created one-sided frequency vectors with a (single-bin wide) carrier at one frequency and two (single-bin) sidebands with fixed and equal distances left and right to the carrier. The carrier has a mangitude of 1 and an angle of 0. The spurs have a magnitude of 0.5 each and can have independent angles.
If I am not mistaken, a $\cos(\omega t) - \sin(\omega t)$ type of expression should have spurs with phase angles of $\pm \frac\pi2$ relative the carrier (coming from the identity $\sin(\omega t) = -\frac{\mathrm{j}}{2}(e^{\mathrm j \omega t} - e^{-\mathrm j \omega t})$. When I configure my spurs with said angles of $\pm \frac\pi2$ and take the inverse fft of my frequency vector, the resulting analytic signal has zero imaginary component and a sinusoidal magnitude. Which means, it is pure AM modulation.
Did I get the phase angles for the $\cos - \sin$ expression wrong?
And what are the general rules here? When I have a carrier (with an angle of 0 w.l.o.g.) and a pair of symmetric spurs with relative angles $\phi_1$ and $\phi_2$, when exactly is the resulting (analytic) signal only AM or only PM?
Edit 2:
Another attempt of mine was to approach this mathematically. Looking at the analytic baseband signal of the demodulated signal from the first edit, it should have an expression of the form $A_1 + A_2 \left(e^{\mathrm j(\omega_s t + \phi_1)} + e^{\mathrm j(-\omega_s t + \phi_2)}\right)$.
As $A_1$ and $A_2$ are constants, I thought I can simply focus on the sum of the exponentials and determine

For which $\phi_1, \phi_2$ is $\frac{\partial}{\partial t}\left|e^{\mathrm j(\omega_s t + \phi_1)} + e^{\mathrm j(-\omega_s t + \phi_2)}\right| = 0$ (no AM)
For which $\phi_1, \phi_2$ is $\frac{\partial}{\partial t}\angle\left(e^{\mathrm j(\omega_s t + \phi_1)} + e^{\mathrm j(-\omega_s t + \phi_2)}\right) = 0$ (no PM)

Unfortunately, for general angles $\phi_1$ and $\phi_2$, I was not able to derive any general conditions for purely AM or purely PM modulation from a single pair of spurs with equal amplitude and distance and independent angles.
Conventional AM and narrowband PM signals look quite similar:
$$x_{AM}(t) = A \cos(\omega_c t) + m(t) \cos(\omega_c t) $$
$$x_{NBPM}(t) = A \cos(\omega_c t) - A k_p m(t) \sin(\omega_c t) $$
And based on this, their spectrum also look quite similar. However, there's an important distinction between them:
AM modulation changes (modulates) the amplitude of the carrier wave, but keeps carrier phase unaltered. Whereas, PM modulation, NW or WB, keeps the amplitude of the carrier constant, but alters (or modulates) its phase according to the message signal.
Ther reason for this different behaviour is about the way the sidelobes (spurs) are added to the main carrier. In AM, the sidelobes are added in-phase with the carrier (using two cosines for that), while in NBPM they are added in quadrature (sine added to cosine). This quadrature additon, leaves the amplitude (almost) unaltered but changes the phase-angle of the carrier.
So if you treat NBPM as an AM and try to demodulate it that way, you will get constant output (or nothing after the DC blocking), as the amplitude of the NBPM carrier is not changing...