Definition of ellipsoid

In chapter 2 section 2.2.2 of Boyd & Vandenberghe's Convex Optimization, the definition of an ellipsoid (which is a convex set) is given as the

$$\mathcal{E} = \left\{x \mid (x - x_c)^{\intercal} P^{-1} (x - x_c) \le 1 \right\}$$

where $$P = P^\intercal$$ is symmetric and positive definite and $$x_c$$ is the center.

If we restrict to real matrices, the way I understand is:

Symmetric real matrices are always diagonalizable, so $$P = RSR^\intercal$$ where $$R$$ is an orthogonal matrix (rotation) with eigenvectors in the columns, and $$S$$ is a scaling matrix composed of the eigenvalues in the diagonals.

So if we multiply all points in the ellipsoid by the inverse of $$P$$, we get a circle (by undoing the rotation and scaling). And the condition for a point $$x$$ to be in the sphere with center $$x_c$$ is that the squared length of $$(x - x_c) \le 1$$. All in all, $$x$$ should be a point in the ellipsoid if

$$x'^\intercal x' \le 1$$

where $$x' = P^{-1}(x - x_c)$$

What am I missing?

Note that $$(AB)^T = B^TA^T$$. So $$x'^T = (x - x_c)^T(P^{-1})^T$$. Thus the correct formula must be
$$(x - x_c)^T(P^{-1})^TP^{-1}(x - x_c) \leq 1$$
But notice that there's no reason you had to call that first matrix $$P$$. Instead, let's call $$P$$ the inverse of that matrix in the middle - so what was "$$(P^{-1})^TP^{-1}$$" is now just called "$$P^{-1}$$".
That also explains why $$P$$ has to be positive-definite!