The shape of the proton-transfer potential between a donor and an acceptor heavy atom is known to depend strongly on the length of the hydrogen bond. It evolves from a double-well to a single-well shape in the limit of very short hydrogen bonds. Ice at high compression is an ideal candidate to study systematically the consequences of this transmutation on properties of hydrogen-bonded solids. We use a computer simulation technique that treats all nuclei as quantum particles and includes, at the same time, the electrons in the framework of density functional theory. Based on this first principles approach we uncover four different regimes. In the Limit of long/weak hydrogen bonds the proton resides covalently bound to a particular heavy atom and zero-point motion broadens its distribution. Ionic defects are shown to play only a minor role. For shorter bonds a regime with strong proton tunneling between donor and acceptor is found. This phase is characterized by a high concentration of ionic defects. At even shorter distances the proton resides midway between donor and acceptor due to zero-point motion, despite the double-well character of the underlying potential. Finally, very short/ultra-strong hydrogen bonds are symmetric because the potential profile is of the single-well type.