Generalized Transfer Function

The damping factor of a complex pair of poles (roots of the characteristic equation) is defined using the the pole position within the complex s-plane. If the angle between the negative-real axis and the pointer to the pole is "alpha" the damping factor d is defined as d=cos(alpha).

It is obvious that cos(alpha) can be expressed by the real part of the pole and the pole frequency (magnitude of the pointer to the pole position).

This definition concerns the complex pole pair only and is, thus, independent on a third pole on the negative-real axis or any zeros of the function.

More than that, the "pole quality factor" Qp is defined as Qp=1/2d.

Hence, for poles in the left half of the s-plane (stable systems) the damping factor varies between "1" and "0" and the pole Qp between "1/2" and "infinite" (oscillation condition).

This is just answering one part of your question:

I don't quite understand the last part. How does he calculate X and ϕ

This is just applying the trigonometric identity $$\sin\left(\alpha + \beta\right)=\sin\alpha\cos\beta + \cos\alpha\sin\beta$$ with \$\alpha=\omega{}t\$ and \$\beta=\phi\$.

Using this identity on the r.h.s. gives $$X\sin\left(\omega{}t+\phi\right)=X\left(\sin{}\omega{}t\cos\phi+\cos\omega{}t\sin\phi\right)$$ so our equation becomes $$\frac{A}{\omega}\sin\left(\omega{}t\right)+B\cos\left(\omega{}t\right)=X\left(\sin{}\omega{}t\cos\phi+\cos\omega{}t\sin\phi\right)$$

Which we can break into two parts, $$\frac{A}{\omega}\sin\left(\omega{}t\right)=X\cos\phi\sin{}\omega{}t$$ and $$B\cos\left(\omega{}t\right)=X\sin\phi\cos\omega{}t$$

So, $$\frac{A}{\omega} = X\cos\phi$$ and $$B=X\sin\phi$$

From there you should be able to get the conclusions from your source.