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.
The power transfer is maximised at resonance because the driving force and the velocity of the oscillator are in phase.
If you multiply two sinusoidal terms together (the force and the velocity) with a phase difference between them, then the product has its maximum average value when the phase difference is zero and a minimum value when the phase difference is $\pm \pi/2$.
Your steady state solution could be correct, but it is more usual to say that if the driving force is $F_0 \sin \omega t$, then the displacement $x \propto \sin(\omega t + \phi)$, where the phase difference $\phi$ is given by
$$ \phi = \tan^{-1}\left(\dfrac{-\gamma \omega}{\omega_0^{2} - \omega^{2}}\right),$$
and $\gamma$ is the damping coefficient.
You can see that when $\omega = \omega_0$ the phase difference between displacement and force is $-\pi/2$. But if you differentiate the displacement to get the velocity $$v \propto \cos(\omega t + \phi) = \sin(\omega t + \phi +\pi/2)$$
and at resonance the phase difference between velocity and force is zero.
If the power transfer is maximised, then this is also why the amplitude is maximised, since the velocity amplitude also increases with the amplitude of the displacement.
Best Answer
Confusion arises because there are a number of ways in which to define resonance.
In mechanical systems, e.g. a spring-mss system, it is much easier to measure amplitudes rather than velocities and so graph to illustrate forced oscillations and resonance are of amplitude of driveN system against frequency of constant amplitude driveR.
When the amplitude of the driveN is a maximum there is amplitude resonance.
The frequency at which amplitude resonance occurs decreases as the damping of the driveN system increases.
For small amount of damping that change in amplitude resonance frequency is small and the amplitude resonance frequency is approximately equal to the free oscillation frequency of the undamped driveN system.
In electrical systems, e.g. an LCR series circuit, current is easier to measure than charge and so it is current which is usually measured to investigated forced oscillation and current resonance.
In that case the maximum current in the driveN system occurs at the same frequency irrespective of the amount of damping of the driveN system.
The current resonant frequency is equal to the free oscillation frequency of the undamped driveN system.
This is also energy resonance where the maximum power is transferred from the driveR to the driveN system.
Velocity resonance for a mechanical system is equivalent to current and energy resonance for an electrical system and charge resonance for an electrical system is equivalent to amplitude resonance for a mechanical system.
Now amplitude $A$ and maximum velocity $v$ are connected $v=\omega A$ where $\omega$ is the frequency of the oscillations.
So the amplitude resonance graph is of amplitude of the driveN $A_{\rm N}$ against frequency of the driveR $\omega_{\rm R}$ whereas velocity resonance graph is of maximum velocity of the driveN $\omega_{\rm R}A_{\rm N}$ against frequency of the driveR $\omega_{\rm R}$.
It perhaps should be of little surprise that the two types of resonance occur at different driveR frequencies.