For some background, see here and here.
Velocity is defined as: “The time rate of change of position of a body; it is a vector quantity having direction as well as magnitude.” Speed is defined as: “The time rate of change of position of a body without regard to direction; in other words, the magnitude of the velocity vector.” (McGraw-Hill Dictionary of Physics, 3rd ed.)
However, it’s not that simple. A common example shows the problem:
When something moves in a circular path (at a constant speed …) and returns to its starting point, its average velocity is zero but its average speed is found by dividing the circumference of the circle by the time taken to move around the circle. This is because the average velocity is calculated by only considering the displacement between the starting and the end points while the average speed considers only the total distance traveled. Wikipedia
So the average speed is not the magnitude of the velocity (which is zero in this case) but something else – the travel distance divided by the travel time.
The question is whether the speed over a finite interval should be the magnitude of the displacement divided by the time interval or the arc length divided by the time interval (i.e., the integral of the norm of the velocity function over the time interval). The answer should be the latter, although the former is implied by the common definition of speed.
It is better to define speed as the ratio of the arc length (length of travel) divided by the arc time (travel time). In short, speed is that which is measured by a speedometer.