  Surfaces and interfaces offer new possibilities for tailoring the many-body
interactions that dominate the electrical and thermal properties of transition
metal oxides. Here, we use the prototypical two-dimensional electron liquid
(2DEL) at the SrTiO$_3$(001) surface to reveal a remarkably complex evolution
of electron-phonon coupling with the tunable carrier density of this system. At
low density, where superconductivity is found in the analogous 2DEL at the
LaAlO$_3$/SrTiO$_3$ interface, our angle-resolved photoemission data show
replica bands separated by 100\,meV from the main bands. This is a hallmark of
a coherent polaronic liquid and implies strong long-range coupling to a single
longitudinal optical phonon mode. In the overdoped regime the preferential
coupling to this mode decreases and the 2DEL undergoes a crossover to a more
conventional metallic state with weaker short-range electron-phonon
interaction. These results place constraints on the theoretical description of
superconductivity and allow for a unified understanding of the transport
properties in SrTiO$_3$-based 2DELs.
