The resonator-induced phase gate is a two-qubit operation in which driving a bus resonator induces a state-dependent phase shift on the qubits equivalent to an effective ZZ interaction. In principle, the dispersive nature of the gate offers flexibility for qubit parameters. However, the drive can cause resonator and qubit leakage, the physics of which cannot be fully captured using either the existing Jaynes-Cummings or Kerr models. In this paper, we adopt an ab initio model based on Josephson nonlinearity for transmon qubits. The ab initio analysis agrees well with the Kerr model in terms of capturing the effective ZZ interaction in the weak-drive dispersive regime. In addition, however, it reveals numerous leakage transitions involving high-excitation qubit states. We analyze the physics behind such novel leakage channels, demonstrate the connection with specific qubit-resonator frequency collisions, and lay out a plan toward device parameter optimization. We show that this type of leakage can be substantially suppressed using very weakly anharmonic transmons. In particular, weaker qubit anharmonicity mitigates both collision density and leakage amplitude, while larger qubit frequency moves the collisions to occur only at large anharmonicity not relevant to experiment. Our work is broadly applicable to the physics of weakly anharmonic transmon qubits coupled to linear resonators. In particular, our analysis confirms and generalizes the measurement-induced state transitions noted in Sank et al. [Phys. Rev. Lett. 117, 190503 (2016)] and lays the groundwork for both strong-drive resonator-induced phase gate implementation and strong-drive dispersive qubit measurement.