I think there is a 2nd interpretation of the impedance matching that
the equivalent impedance from the output of the box to the RF
generator becomes the complex conjugate of ZL. This impedance is the
impedance seen by the ZL when it looks back toward the RF generator.
I'm wondering whether or not it is right.
You're right, although it's not a "2nd interpretation", but a simultaneous condition fulfilled by the matching network. A proper 2-port impedance matching network matches both ports: the one looking towards the generator and the one looking towards the load.
Also, the condition for matching always leads to "becoming" the complex conjugate of the impedance you're looking into. In the case of the generator the complex conjugate requirement drops off because its source impedance is purely resistive (no imaginary part, so no complex conjugation applies).
Once the power transfers the ZL, the power propagates from the one end
of the cable to ZP. However, it is general that the ZP is not equal to
Z0, so some power is reflected from ZP and propagates back.
Wrong (although true)!
Those reflections are there indeed (and they are infinite in number), but they are only relevant for a time domain (transient) analysis. When enough time has passed and steady-state has been reached, all those multiple reflections sum up and can be simplified down to just one travelling wave. If the reflections happen to sum up with certain phase differences so that they cancel each other, the net result in steady-state is zero. That's why we say that there is no reflection, which is a language abuse because in fact what is happening is that there is no effective reflection (no effective power is reflected).
So in this case forget about time-domain reflections and bear with the steady-state frequency-domain framework. Let's see what happens when we analyse from this point of view.
First things first: \$Z_0\$ is just the characteristic impedance of the cable, but that doesn't mean that the plasma reactor is seeing a \$Z_0\$ impedance. In fact, if impedance matching has been properly done, the reactor should be looking into a \$Z_{out}^{\prime} = Z_p^* \neq Z_0\$ impedance.
Note that the characteristic impedance isn't an actual impedance! It's just a parameter of a transmission line. This is a very common pitfall. Keep it in mind to avoid falling into it again.
Then, what's the actual impedance \$Z_{out}^{\prime}\$ seen by the plasma reactor? Well, it depends on the cable length \$l\$, the output impedance \$Z_{out}\$ of the LCC matching network and the cable parameters \$Z_0\$ and \$\beta\$. The relationship is as follows:
$$
Z_{out}^{\prime} = Z_0 \cdot \frac{Z_{out} + j Z_0 \tan{\beta l}}{Z_0 + j Z_{out} \tan{\beta l}}
$$
As you can see for yourself, in general \$Z_{out}^{\prime} \neq Z_0\$. The only case when \$Z_{out}^{\prime} = Z_0\$ is when \$Z_{out} = Z_0\$, which is not going to be your case because your plasma reactor is mismatched.
It's worth mentioning that there is a particular case that simplifies dealing with that cable. When the length of the cable is equal to \$n \frac{\lambda}{2}\$ (n is an integer 1, 2, 3...) then \$\tan{\beta l} = 0\$ and:
$$
Z_{out}^{\prime} = Z_{out}
$$
That is, in that case you can ignore the cable because it has no effect on the impedance seen by the plasma reactor... it's like the reactor is connected directly to the LCC matching network.
You might want to take a look on the theoretical foundations of transmission lines to better understand what's going on in your cable. In that case, I'd recommend you to read this, and focus on sections 11.6 to 11.9 and 11.14.
Since Z0 is generally also not equal to the impedance seen by ZL
looking toward the generator, some power also reflects at the
interface between the cable and the matching box output and the rest
power transmits back to the matching box.
Also wrong (although true) ! Same principle as above applies here.
So... there is a power reflection toward the generator even if the
impedance matching is done?
No, there shouldn't be any steady-state reflections if impedance matching is done properly. That is only accomplished if you take into account the effect of the matching box AND the cables.
You should think of the 2 cables as if they where just displacing the reference planes of the matching box ports. In other words, you should embed the effects of the cables into an "equivalent matching box".
Best Answer
Reflections occur or are noticeable when there is a transmission line involved and that transmission line is long enough for significant reflections to occur. This is generally accepted to be a length of about one-tenth of a wavelength. So, at 1 MHz, the wavelength is 300 metres and so unmatched transmission line problems start at about 30 metres. Higher frequencies naturally have unmatched problems on shorter line lengths.
However, the impedance of a transmission line for radio frequencies of about 1 MHz and above can be taken to be purely resistive. In other words it doesn't present a complex impedance hence it should be matched with an equivalent resistance to avoid reflection problems and this also ties in with the maximum power transfer. So no real problems here.
For an antenna, it can have a highly capacitive impedance if it is regarded as "short". An example being a monopole that is less than one-quarter of a wavelength. The radiation resistance it would naturally present when a quarter wave long would fall from 37 ohms to a much smaller figure when the antenna is shortened. The effective series capacitive reactance rises from near-zero at a quarter wave to tens, hundreds or thousands of ohms as the antenna shortens.
So this is an example of where using an inductor (a conjugate component) can cancel the short antenna's capacitive impedance and allow a better transfer of power.
Of course there is a reflection - that is the mechanism by which we get an impedance transformation to that of free space at a particular frequency. And, adding a conjugate component to cancel out the inherent capacitive reactance of a "short" antenna doesn't alter how the antenna works but it does allow a better transfer of power.