A slope can be positive, negative, zero or undefined, but what about the magnitude of the slope? It seems useful to recognize if the slope is steep (greater than $1$) or gentle (less than $1$) as much as it is positive or negative. Those words don't seem to be used in basic algebra, though.
Is there a mathematical term for telling if $|m|$ is greater than or less than one?
On
There is nothing special about your arbitrary choice of the absolute value of the slope being less than 1.0, and no reason whatsoever for a special term for it.
You're looking for a new term related to a range of derivatives, so you might be interested to learn that the subderivative of a function at a point $x$ on a one-dimensional function (even one with discontinuous traditional derivative) is the range of derivatives consistent with limits at that point. Thus the subderivative at $x=0$ for this blue function is range $[-1, 1/2]$. That is, you can draw a line through the point at $x=0$ with any slope in the range $[-1, 1/2]$ and it will touch $f(x)$ only at $x=0$.
The subgradient is the multi-dimensional generalization.
There is not any particular term but observe that $m=\tan \alpha = 1$ for $\alpha = 45°$ thus we can say that $|m|>1$ when the angle to go up or down is greater than $45°$.