You are reading into the rule more than is there. The rule simply addresses when the defendant may waive his right to a jury trial. It does not on its face directly address whether the state has a right to a jury in a criminal case when the defendant has waived his right. Allowing a defendant to waive his right to a jury trial is not the same thing as saying the defendant has a right to a bench trial.
That said, you happen to be correct that in Ohio the defendant has the right to a bench trial if he/she wants one. Some legislators in 2011 tried to change Ohio law to conform to the law in the federal system and the majority of states that effectively gives the prosecutor a right to a jury. In a legislative analysis provided to the legislature by the Ohio Judicial Conference (OJC), which speaks for the Court system, the OJC stated:
House Bill 265 proposes to condition the defendant’s ability to waive a jury trial on the consent of the court and the prosecuting attorney. Twenty-nine states, the Federal criminal justice system, and Washington D.C. require the consent of the court and the prosecuting attorney in cases where a criminal defendant wishes to waive their right to trial by jury. This is a significant shift from the long standing policy in Ohio that the defendant alone has the right to demand a jury trial.
If you read the analysis I linked, you will see that the OJC was very much against the idea of changing Ohio law to allow the state a right to a jury trial. But in the federal system and the majority of states, the prosecutor does have that right.

