Could you explain what this means by "Doesn't minimize the correct error function"? How is the least-squares error that we minimized not "correct"?

This is basically the same question, but how is the error function between the projected 3D points and the image positions (E) different?

So check out this slide. We derived a set of constraints used for performing our geometric camera calibration:

$$x' = \frac{p_1^T X}{p_3^T X}, y' = \frac{p_2^T X}{p_3^T X}$$

Our "ideal" error function $E_1$ between image positions and projected 3D points involves minimizing Euclidean distance in 2D image space:

$$E_1(p_1,p_2,p_3) = \sum_k\left(x_k' - \frac{p_1^T X_k}{p_3^T X_k}\right)^2 + \left(y_k' - \frac{p_2^T X_k}{p_3^T X_k}\right)^2$$

Unfortunately, minimizing this objective requires a nonlinear optimization procedure.

Instead, as described in this slide, we can linearize our system of equations:

$$x' p_3^T X - p_1^T X = 0, y' p_3^T X - p_2^T X = 0$$

Using a least squares procedure, we effectively end up minimizing an objective with the following terms:

$$E_2(p_1,p_2,p_3) = \sum_k(x_k'p_3^T X_k - p_1^T X_k)^2 + (y_k'p_3^T X_k - p_2^T X_k)^2$$

While this is easier to solve (i.e., it has an analytical solution), $E_1 \neq E_2$.