terminology - How do you define the resonance frequency of a forced damped oscillator?

Consider a forced, damped harmonic oscillator

$$\ddot{\phi} + 2\beta \dot{\phi} + \omega_0^2 \phi = j(t) \, .\tag{1}$$

If I pick a sinusoidal driving force $j(t) = A \cos(\Omega t)$, I find

$$\phi(t) = \text{Re} \left[ e^{-i \Omega t} \frac{-A}{\Omega^2 - \omega_0^2 + 2i\beta \Omega} \right] \, .\tag{2}$$

From here, how do I define the "resonance"? Is it the point where $\langle \phi(t)^2 \rangle$ is maximized?

Things I do know: The frequency at which $\langle \phi(t)^2 \rangle$ is maximized is $$\omega_r ~:=~ \omega_0 \sqrt{1 - 2(\beta/\omega_0)^2},\tag{3}$$ but I thought I read/heard that the resonance frequency of a damped oscillator is just $\omega_0$.

I also calculated that the free oscillation frequency is $$\omega_{\text{free}} ~:=~ \omega_0 \sqrt{1 - (\beta / \omega_0)^2},\tag{4}$$ but I don't think that's the same thing as the resonance frequency under steady driving.

Answer

From here, how do I define the "resonance"?

At resonance, the energy flow from the driving source is unidirectional, i.e., the system absorbs power over the entire cycle.

When $\Omega = \omega_0$, we have

$$\phi(t) = \frac{A}{2\beta \omega_0}\sin\omega_0 t$$

thus

$$\dot \phi(t) = \frac{A}{2\beta}\cos\omega_0 t$$

The power $P$ per unit mass delivered by the driving force is then

$$\frac{P}{m} = j(t) \cdot \dot \phi(t) = \frac{A^2}{2\beta}\cos^2\omega_0 t = \frac{A^2}{4\beta}\left[1 + \cos 2\omega_0 t \right] \ge 0$$

When $\Omega \ne \omega_0$ the power will be negative over a part of the cycle when the system does work on the source.

What you've labelled as $\omega_r$ is the damped resonance frequency or resonance peak frequency.

Unqualified, the term resonance frequency usually refers to $\omega_0$, the undamped resonance frequency or undamped natural frequency.