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.
 A: Confusion might arise because the use of the word resonance often differs between mechanical systems and electrical systems.  
With mechanical systems it is often the case that displacement resonance is considered and the frequency of displacement resonance decreases as the amount of damping increases.
This is the frequency dependence stated in your equation (3).  
When it comes to electrical systems eg a series LCR circuit the parameter which is often measured is the current and the frequency at which current resonance occurs is the natural frequency of undamped oscillations of the system $\omega_0$ and as such the frequency for current resonance does not vary with damping.  
For a mechanical system current resonance is velocity and power resonance and for an electrical series LCR system displacement resonance is charge resonance.  
In science and engineering courses where mechanical and electrical forced oscillations are first discussed displacement resonance is favoured for some mechanical systems because it is easier to measure a distance than a speed and current resonance is favoured for some electrical systems because it is easier to measure a current than a charge.
A: 
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.
