In this work, we analytically compare the performance of the time- and unit cell-splitting protocols for satisfying the energy needs of reconfigurable intelligent surfaces (RISs) through wireless energy harvesting from information signals. We first compute the RIS energy consumption per frame for both protocols and subsequently formulate an optimization problem that maximizes the average rate under the constraint of meeting the RIS long-term energy consumption demands. Analytical solutions to the optimal allocation of resources that involve a single integral are provided for both protocols in the case of random transmitter-RIS links that are subject to Rician or Nakagami-m fading distributions. Moreover, closed-form solutions are provided for the case of deterministic transmitter-RIS links. In addition, increasing and decreasing monotonic trends are revealed, based on analysis, for the ratio of the achievable rates of the presented protocols with respect to the RIS energy consumption. Finally, numerical results validate the analytical findings and reveal that the unit cell-splitting protocol exhibits a notably higher average rate performance compared with its time-splitting counterpart throughout the feasible range of RIS energy consumption values. However, this comes at a cost of a notably reduced signal-to-noise ratio as the RIS energy demands increase.