Linguistically speaking, in the U.S. the Presidency is just a job like any other. We do not use any special word, and we most certainly avoid kingly words like "rule" or "preside". We might say he "serves his term". You can use the same words for any job. For example, you could say that a person who delivers mail for the Postal Service "has served for five years delivering mail".

Preside doesn’t fully capture all the duties of the president, especially outside formal sessions. The most natural verb for a president would depend on the context of their role: lead - Let the president lead. (most common and captures overall responsibility) govern - Let the president govern. (fits political or national leadership) direct - Let the president direct. (common in organizations or meetings) oversee - Let the president oversee. (emphasizes supervision) rule - Let the president rule. (more authoritative, often used in historical or monarchic contexts)