Agreeing that the definition of a “song” is a piece of music with lyrics, we should surely have a type for a piece of music that has no lyrics, but does not fall into any of the other choices?
I propose a new type “Instrumental” to be added, as per this wikipedia article https://en.wikipedia.org/wiki/Instrumental
An instrumental is a recording without any vocals, although it might include some inarticulate vocals, such as shouted backup vocals in a Big Band setting. Through semantic widening, a broader sense of the word song may refer to instrumentals. The music is primarily or exclusively produced using musical instruments. An instrumental can exist in music notation, after it is written by a composer; in the mind of the composer (especially in cases where the composer themselves will perform the piece, as in the case of a blues solo guitarist or a folk music fiddle player); as a piece that is performed live by a single instrumentalist or a musical ensemble, which could range in components from a duo or trio to a large Big Band, concert band or orchestra.