I'll take a stab at this, but I'm NOT an authority.

My understanding of the term "air" is that it means a melody or tune, and is used in reference both to the melodies of songs which also have lyrics as well as to purely instrumental tunes.

In fact, I believe that the common use of the word "tune" in Anglo/Celtic folk-music circles as referring specifically to instrumental pieces without vocal parts is why the alternative term "air" is used in reference to melodies considered on their own, whether or not they are customarily performed with or without words.

Some people, if not all of them, use the word "waltz" to refer to anything in 3/4 time. I would be interested in learning what criteria, if any, might be used to differentiate "true" waltzes from other 3/4-time pieces.