QuasiUnaFantasia I think method 1, as listed, is quite primitive synthesis, as found in 1980's synthesizers, using a limited number of parameters to tweak basic wave patterns into shape. Method 4 is mathematically similar, it's just that the parameters have physical meaning, in terms of piano's, whereas the primitive methods use parameters withouth any meaning in a piano context.
I think differently.
Method 1 is similar to the ole-fashioned synths, but it can be extended at will. Back then, you had 3 waveforms, one envelope and that was it. Nobody prevents you to make it as sophisticated as you wish, with direct harmonics computation (rather than fixed waveforms) and with more complicated envelopes, which can vary from harmonic-to-harmonic in one note (as it actually happens on an acoustic piano, since some harmonics decay faster than other ones which "bloom"). Both the harmonic content and the envelope can vary also from note to note and within the same note from velocity to velocity. Still the same principles. My understanding is that Pianoteq does this, and see below for yet another hint of why that's the case.
On the other hand, Method 4 is solving the partial differential equations (PDE) for the physical model of the piano. Most typically this is done in the time (not frequency) domain, even though some spectral element solvers do exist. You don't do any "sound" simulation per se, you just model the dynamics of the instrument. For this model, you need the detailed geometry of the instrument, the detailed mechanical characteristics (e.g. various materials Young's modulus, string tension, impedance of the various transition points, e.g. string-bridge-soundboard, soundboard-rim etc). In this model, there is no way that you could "manually" increase or decrease some harmonics (as Pianoteq allows one to do) without completely breaking the PDE solvers, whether it's in the time or frequency domain.
So the fact that PTQ allow you to change the harmonics, and that it does not ask you for any of the physical parameters which I would expect (besides string length) leads me to believe that they do Method 1, perhaps augmented by some "proxies" for the physical properties of the instrument itself.
This is further reinforced by the first (at time of writing) comment on the video linked below and by similar statements all over Modartt website: they use recordings of pianos trying to match the sound. If their approach was all-out for 4 this would be simply impossible, and on the other hand they would be very carefully measuring the geometry of the instrument and its mechanical characteristics, which they never said or wrote anywhere that they do. Heck even the string lengths of many of the instruments they have do not match the instrument they are claiming to model!!!
Even more explicit are the demos on their website at https://www.modartt.com/pianoteq#pane_technology which they exactly go from sine waves to the final "modeled" piano sound with the "advanced synth" approach I described above, rather than in the "solve the equation of the motion of the strings, soundboard and air" which are needed for Method 4.
So in my opinion this is strong evidence for PTQ utilizing Method 1, as I said perhaps augmented with some small details based in physics, but certainly not a full-fledged physical modeling.
https://www.youtube.com/watch?v=vm2-WPUr6P8
