Viewing the Milky Way

Okay, a general question: When people on earth view the night sky and
say, "There's the milky way"; isn't that so redundant, it's wrong??? I
mean we're IN the milky way and can see nothing but the milky way
(save for a few M31, Andromeda extragalactic glimpses), and when we
see a band of the milky way in the night sky; that's really just an
arm of the milky way's spiral, right? (is that the orion or the
sagitarius arm?) so what we really should be saying is the "there's an
arm of the milky way"; saying "there's the milky way" from earth, can
be equated to being in a room, looking at a wall and saying, "oh,
there's the room!" (what should be said is there's a WALL of the room)
Right?? People have pointed out the milky way in the night sky
multiple times and that always confused me, but I've finally wanted to
clarify this that their phraseology was incredibly misdirected.

Let me know if this is on the right track or not. seems like I've got
the gist of it. thanks.