He wrote these pieces to instruct his oldest and highly gifted son, Wilhelm Friedemann, in the art of cantabile composition. Now, to be honest, 40 years ago I would not have had a clue what that might be. I think I now have an idea.
Cantabile during Bach's day would correspond to moderato for Haydn or Schubert, or rubato for Chopin, which is to say, music in which the beat itself is not strictly measured.
In order to play this way one must know when a note is not a note, but a reference to another kind of musical gesture: a sigh, perhaps, or a yodel.
Did you or anyone you know ever learn a Bach Two-Part Invention as anything other than strictly strict?