Using rate and per unit time together for describing effects of motion

No, using "rate" and "with respect to time" together is correct and encouraged.

A "rate" is a change in one quantity with respect to another quantity. You can have quantities that change over time, like the concentration of products in a chemical reaction changing over time. But you can also have quantities that change over space, like the barometric pressure changing across the country. You can also have quantities that change with respect to other quantities, like the pressure changing with respect to temperature.

All of these are expressed mathematically with a derivative: $$frac{dy}{dx}$$ where $y$ is the quantity that is changing and $x$ is the quantity it is changing with respect to. So then velocity is $dvec r/dt$ where $vec r$ is the position and $t$ is time. The change in barometric pressure across the country would be $dP/dx$ and the change in pressure with respect to temperature would be $dP/dT$.

The best way to say $dvec r/dt$ in words is "rate of change in position with respect to time". The phrase "rate of change of position" is acceptable as long as "with respect to time" is understood from context. Since rates of change are not always with respect to time it is not unnecessary to state what the change is with respect to.