Crystal compensation for temp and voltage
AT-cut crystals may have 3rd-order temperature profiles as outlined by OP, but tuning-fork 32768 crystals most often have a far different 2nd-order temperature profile, something like:


from IQD datasheet LFXTAL062558Reel.pdf



In addition, these tuning-fork crystals should be driven with much less power than AT-cut crystals. Resonant drive level might be 0.1 uW versus up to 100 uW for AT-cut. If you overdrive a crystal, frequency stability is degraded and spurious crystal resonances may take over.

A low drive level means oscillator noise power may become important.