"Since I've seen that galaxies are often called "deep sky objects", as opposed to individual stars, does this mean that all visible stars in the night sky actually only belong to The Milky Way Galaxy?"
-When you look up at the night sky, the many objects you see could be stars, galaxies, quasars, nebulas, and many other massive bright objects in the sky. Even through they look like "stars" they may not be. Some of these objects may belong to the milky way (stars, meterors, nebulas, planets etc) but there are many objects that we see that are much further from the outside of our galaxy.
For example: With your eye during certain times in the year, you can actually see the Andromeda Galaxy from Earth! It may appear as a "star" like object in the sky, but in fact it's a galaxy much like our own!
"Or are there stars which belong at least to the galaxies in the Local Group? I always thought that Milky Way was that lighter band (since it is said that the Galaxy is a "disc"), and that other stars might be part of other galaxies."
Some of the objects in the night sky that are outside of our Milky Way may be apart of the local group....they may not be. They could be apart of some other group of galaxies etc. It all depends on how massive they are, and how much light they emit. This will determine whether or not we can see them from Earth!