I was told that the name 'Milky Way' first came about when people observered the milky looking stretch of sky; that is, before telescopes came about and we realised it wasn't a milky cloud, but rather a collection of stars, our galaxy.
So strictly speaking, Milky Way refers to that visible milky patch in the sky, rather than the collection of stars that is our galaxy, and that over time they've come to have the same name.
Is this the case? If so, does our Galaxy itself have its own name?
As far as I know the name of the galaxy is "The Milky Way Galaxy". The evolution of the name is as you describe but it has come to be used as the name of the galaxy as well not just what is visible on the sky.
"The Milky Way Galaxy" is indeed the correct name for our Galaxy in western culture, the name "Milky Way" comes from the Latin Via Lactea (translated from Greek).
The name varies from culture to culture, for example the Chinese use "Silver River" to represent our Galaxy.
Wikipedia has a good list of name for the Milky Way.
I think the important thing, as Patrick brought out, is that the name "Milky Way" is what the galaxy is called in Western culture. Names change between cultures and between languages. As far as astronomical objects go, often they will have ties to ancient cultures (e.g. Mercury, Jupiter). In English, the name of our galaxy is The Milky Way Galaxy. Sometimes, it is even referred to simply as The Galaxy.
This same principle applies to many other objects: The Moon, The Sun, The Solar System. Those are the actual names of those objects (or systems).
Although The Milky Way is thought to be our galaxy there is also a confusion that we have actually colided with The Milky Way and we are actually part of a much smaller "dwarf galaxy" called The Sagittarius Dwarf galaxy however this is apparently just some internet rumor. You can read an article about it here.
