We use ab initio molecular dynamics simulations based on density-functional theory to revisit the enigmatic capacitance peak of the electrified Pt(111)/water interface around the potential of zero charge. We demonstrate that counterbalancing the electronic excess charges with partially charged hydrogen atoms constitutes a computationally efficient approach to converged interfacial water structures. The thus enabled detailed analysis of the interfacial water response clarifies that the peak in the capacitance is predominantly due to structural reorientation, although its magnitude is significantly increased by strong internal electronic polarization, also known as charge transfer (CT). We find that CT is more complex than previously thought, resulting from the interplay between chemisorbed water and depolarization effects from the surrounding water. Finally, we demonstrate that quantitative agreement with the experimental peak can be achieved through inclusion of the interfacial response into an implicit solvent model for the extended part of the double layer. This suggests that such models can accurately reproduce screened interfacial fields as a function of potential, despite their notoriously small native capacitance.