Your dish is not like an aerial. It receives the satellite signal and tunes in the selected channel. On some receivers, and satellite PVRs, you can tune in to two channels. That is it. There is no output from the box that carries all 300 channels that you can lead somewhere else and select stations from there (not legally anyway). Your one or two channels is it.
When you buy your service, an authorization code is loaded into your receiver and your receiver also gets the list of channels you have paid for. The latter is sent by satellite so if you add or remove channels the receiver is updated.
To connect to your PC, you can use the HDMI, DVI, 75ohm, component video, s-video, composite video, USB but it will only carry that one channel. To change channels, you do so with your satellite receiver remote. That is your tuner.
With a 75ohm aerial, you can use a TV tuner to select and view terrestrial channels. With a cable box, there will be the same sort of authorization as with satellite. Like satellite, the cable box is often your tuner too.