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

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.