Different Ways to Say "Hometown"
To me, either of 老家, 家乡, or 故乡 gives similar meaning, any of one doesn't not necessarily mean the home that I'm living in currently, nor other implications. Context can determine the implied meaning. All of them can be used as written expression, and also can be heard often. 故乡 seems to be the most formal among these three.

Another one, 祖籍 (or 籍贯), is quite definite: home of my ancestor, usually being a different one from my current home.