does this mean that all visible stars in the night sky actually only belong to The Milky Way Galaxy?
Yes. The only naked eye objects not part of the Milky Way Galaxy are the three or four galaxies visible to the naked eye: the two Magellanic Clouds, the Andromeda Galaxy, and possibly the Triangulum Galaxy.
Or are there stars which belong at least to the galaxies in the Local Group?
No, all the stars in other galaxies are too faint to be visible with the naked eye, with the brief exception of the supernova in the Large Magellanic Cloud, visible in 1987.
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.
The term "Milky Way" refers to two different things. Traditionally it referred to the pale milky band around the night sky, the result of the combined glow of millions of stars too faint to be seen with the naked eye. In the last century or so it has also been used to refer to our local Galaxy, which includes both this band and also all the stars in the sky, which are all in our particular region of the Milky Way Galaxy.