Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

Subtracting $x/2$, and rescaling, the problem is reduced to the best uniform approximation of $|x|$ on $[-1,1]$ by polynomials, a problem already considered by Bernstein.
According to Chebyshev's theory, the polynomial of degree $\le n$ that minimizes the uniform distance on $[-1,1]$ from the absolute value function is unique and is characterized by Chebyshev's equioscillation theorem. It is also even, by symmetrization argument. More details should be easily available in the literature; for instance, these notes devote the whole chapter 3 exactly to the problem.

This is done by using Remes's algorithm. It is implemented in Mathematica's command MiniMaxApproximation[]. Below is an image of the corresponding Mathematica notebook, giving a polynomial approximation of degree $10$. (Click on the image to see it better.) The quality of the approximation seems substantially better than in your picture.