A map $f: X \to Y$ is a quasi-isometric embedding if there exists constants $K \ge 1$ and $C\ge 0$ so that for all $x, x' \in X$ we have $$\frac{1}{K}d_X(x,x') -C \le d_Y(f(x), f(x')) \le K d_X(x,x') +C.$$
But we could also let $R=\max\{K,C\}$ and the quasi-isometric embedding still holds.
Why is this not used as the definition?