Games and Scott sentences for positive distances between metric structures

We develop various Ehrenfeucht-Fra\"{\i}ss\'{e} games for distances between metric structures. We study two forms of distances: pseudometrics stemming from mapping spaces onto each other with some form of approximate isomorphism, and metrics stemming from measuring the distances between two spaces isometrically embedded into a third space. Using an infinitary version of Henson's positive bounded logic with approximations, we form Scott sentences capturing fixed distances to a given space. The Scott sentences of separable spaces are in $\mathcal{L}_{\omega_1\omega}$ for 0-distances and in $\mathcal{L}_{\omega_2\omega}$ for positive distances.
Comment: This is a generalization of the previous version, based on ideas inspired by other papers on the subject and referee feedback