The interest rate itself rises

Money demand stays put. Money supply stays put.

The interest rate is the variable on the vertical axis. A change in it is a movement along the money demand curve, not a shift of it. At a higher interest rate, people choose to hold less money because holding cash forgoes more interest. Neither curve moves. If something did shift, the stem would have named it.

The quantity of money demanded falls along the unchanged money demand curve, and neither curve moves.

Common mistake. The classic slip is to shift money demand left to 'show people holding less money.' They do hold less, but that is a movement along the curve. Only changes in income, prices, or payment technology shift money demand.