Wikipedia says:

In relativity theory, proper acceleration[1] is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object.

and says:

In the standard inertial coordinates of special relativity, for unidirectional motion, proper acceleration is the rate of change of proper velocity with respect to coordinate time.

Why coordinate time?

As far as I know the moving observer itself measures the proper things. It's clock measures proper time, it's speedometer measures proper velocity.

So if the observer accelerates with $10m/s^2$ (and feels that proper acceleration). Isn't the speed shown by the local speedometer changes $10m/s$ per second on the local clock?