  We consider the quasi-classical limit of Nelson-type regularized polaron
models describing a particle interacting with a quantized bosonic field. We
break translation-invariance by adding an attractive external potential
decaying at infinity, acting on the particle. In the strong coupling limit
where the field behaves classically we prove that the model's energy
quasi-minimizers strongly converge to ground states of the limiting Pekar-like
non-linear model. This holds for arbitrarily small external attractive
potentials, hence this binding is fully due to the interaction with the bosonic
field. We use a new approach to the construction of quasi-classical measures to
revisit energy convergence, and a localization method in a
concentration-compactness type argument to obtain convergence of states.
