This makes sense to me, because, presumably, water is vaporising until the pressure rises to the point that it is sufficient to contain/compress the remainder of the water in liquid form.
I suggest you don't think of it as "containing" the water. Instead, think of it that there are 2 processes happening simultaneously: evaporation and condensation.
Given the vapor pressure of the liquid, we know that some amount of the liquid is constantly being turned into gas/vapor. If there is minimal vapor in the chamber, then this evaporative process puts more vapor in.
When the partial pressure of the vapor in the chamber equals the vapor pressure of the liquid, it's not the case that everything stops or that the liquid is contained. Instead the two processes are simply happening at the same rate. Water is still vaporizing, but just as much is condensing. (There is also the case that evaporation is favored for smaller droplets, so in the equilibrium condition, tiny droplets don't form).
If we increase the partial pressure of the vapor by compressing the volume, the rate of condensation increases and is now greater than evaporation. Any droplets that form will increase in size since deposition is faster than removal.
So the process of condensation doesn't start when you compress the cylinder, it's happening all the time. The only thing that changes is the rate of condensation.
But isn't this "containing" interpretation necessary to understand the the idea of a "compressed liquid" i.e the reason something is a liquid at a particular temperature is that the pressure is large enough to keep it as a liquid
If you like, that's fine. I'm just trying to emphasize that both evaporation and condensation are happening all the time. There's nothing special that happens to the liquid at and above the vapor pressure. The liquid is evaporating both above and below that point. It's just that above the vapor pressure, it's dominated by condensation, so the net volume of the liquid is increasing.
In the end you say, higher partial pressure (of the vapor), greater the condensation. But if instead of pushing the piston down, if I introduce some air in, that would lead to more condensation of the water vapour also. In this case I have changed the total pressure, keeping the partial pressure of the vapor the same. So doesn't this mean condensation depends on total pressure
That should not happen. If you introduce pure nitrogen, you're increasing the total pressure (so you might stop the liquid from boiling). But if you're not increasing the partial pressure of the vapor, then you're not changing the condensation rate. The liquid will still have a net evaporation, driving the partial pressure higher toward the vapor pressure of the liquid.
"air" of course may have some water vapor in it. If compressed, it may be above or below the vapor pressure of the water in the chamber.