The frame of reference in which the mountain peak is stationary is not inertial -- due to the Earth's rotation, it's actually constantly accelerating toward the center of the Earth. Hence, the quote on special relativity you posted doesn't (directly) apply.

However, time passes slower the deeper one gets into a gravity well. The center of the Earth is the center of Earth's gravity well, so the closer to it you are, the slower your clock ticks.

More info: https://en.wikipedia.org/wiki/Gravitational_time_dilation