When and how does one use a Wiener filter?
Perhaps a motivating radar/audio example would be an adaptive sidelobe canceller and adaptive noise cancellers. Rather than just show equations, let's walk through some text descriptions:

Let's say you are trying to record someone singing, but you're in a large auditorium. Your microphone is picking up the singer, but it's also picking up a lot of the reverberation. Now if we can characterize the reverberation somehow, we'll be able to remove it from our signal of the singer+reverberation. Enter another microphone (or a group of microphones), set up to just record the reverberation. With this setup, we can construct an adaptive filter that will remove the unwanted reverberation.

Another example would be some basic electronic protection for a radar system, where we want to remove jamming interference. In a radar system, you have a main channel which extracts the signal you're interested in. Sometimes, the case of some electromagnetic interference that originates from outside of the main antenna beam, i.e. from an antenna sidelobe, arises. This severely corrupts our main channel with noise, and may obscure target returns. 

Intuitively, if we can somehow characterize just the interference, we can construct an adaptive filter that will remove the noise from the main channel. We can do this using single elements with isotopic (or nearly isotopic) responses. These auxiliary channels as they're called have very low gain compared to the entire antenna, and thus will receive only the interference and not a target which may be in the main beam. 

The downside with wiener filters is that such a solution requires the noise to be stationary. Fortunately, we can use filter banks, time segmentation, and other signal processing techniques to make that possible.

Perhaps someone with a communication systems background could chime in and talk about adaptive equalizers, that's another application where you're trying to characterize some noisy channel, and you transmit a training signal, which the system knows about. The idea then is basically your first example: you know what you sent out, and you get some noisy version back. Using an adaptive filter, you can characterize that noise and now be able to remove it for anything else you send out (assuming the noise is perfectly stationary).

Hopefully that gives you some insight and motivation!