And actually, wouldn't nearly _everything_ you see be the Milky Way galaxy? That's not the point of the site, but struck me just now. I had the same experience you did, while backpacking in New Mexico. The sky was outrageous - I had never seen the whole of the structure.
Yes, pretty much everything visible to the naked eye is going to be the Milky Way. There are a few deep sky objects like the Andromeda Galaxy that you can see from a reasonably dark site though, if your eyesight is good and you know right where to look. (It's pretty easy to find with binoculars but with the naked eye it will just be a small, faint, diffuse glow that doesn't look particularly interesting, assuming you can see it at all.)
In an absolutely trivial sense, the earth, and everything on it, is part of the Milky Way. All that light pollution is light from the [0]Milky Way reflecting off the [1]Milky Way and thereby obscuring the [2]Milky Way from the [3]Milky Way.
[0] cities on a planet of the
[1] atmosphere of a planet of the
[2] majority of the stars of the
[3] only known inhabitants of the
Sorry to sound silly, but I've only seen real milky way in Americas, but never in Asia. It's a common knowledge to me that you can see it, but it doesn't quite match the reality for me.