$v$ isn't referring to either $v_1$ or $v_2$, necessarily; $v_1$ is representing the vector before it moves, and $v_2$ is the vector after this movement. If we are working in polar coordinates (the reason he is using $v_\perp$ and $v_\parallel$), then let's suppose this small movement isn't changing the magnitude of the vector, it is just changing the direction (hence why we just care about $v_\perp$).
In very small rotations, we can approximate the distance moved in the rotation simply by the magnitude of the vector multiplied by the angle moved. Hence we get
$$\Delta v_\perp\approx v\Delta\theta$$
In other words, look at figure 11-8. Imagine that the vector $v_\perp$ is very small. Then you will notice that $v_1$ and $v_2$ will converge to the same length, we call this $v$. Then try to see if you can convince yourself that what I said before is true.