Understanding JWST water spectra: what can thermochemical models tell us about the (cold) water in protoplanetary disks?

(Abridged) Rotational H$_2$O spectra as observed with JWST/MIRI provide a good probe of the temperature and column density structure of the inner disk. H$_2$O emission can also be influenced by dynamical processes, such as dust grains drifting inwards and their icy mantles sublimating once they cross the snowlines, thus enriching the inner regions in H$_2$O vapor. Recent work has found that this process may leave an imprint in the H$_2$O spectrum in the form of excess flux in the cold H$_2$O lines. In this work, we aim to test the accuracy of several common retrieval techniques on full 2D thermochemical disk models. Moreover, we investigate the cold H$_2$O emission that has been proposed as a signature of drift, to gain further insights into the underlying radial and vertical distribution of H$_2$O. We present two sets of Dust And LInes (DALI) thermochemical models and run several retrieval techniques to investigate how the retrieved temperature and column density compare to our models. Single-temperature slab retrievals mainly trace the warm ($\sim$500 K) H$_2$O reservoir, whereas a three-component fit is able to better trace the full temperature gradient in the IR emitting region. Retrieved temperatures tend to underestimate the true temperature of the emitting layer due to non-LTE effects. The retrieved column density traces close to the mid-IR dust $τ=1$ surface. We find that the strength of the cold H$_2$O emission is directly linked to the H$_2$O abundance above the snow surface at large radii (>1 au). This implies that sources with excess cold H$_2$O flux likely have a high H$_2$O abundance in this region ($\gtrsim10^{-5}$), higher than predicted by the chemical network. This discrepancy is most likely caused by the absence of dust transport processes in our models, further strengthening the theory that this emission may be a signature of radial drift and vertical mixing.