Melodist: Aquinas argues in the quoted portion that the Son proceeds by way of intellect (the logos) while the Spirit proceeds by way of will (love). Would this be the distinction you were asking for?
I think you might want to re-examine the part you said makes sense (not that re-examining would make you agree with it, but I think maybe you missed why you would disagree with it, as it seems on my initial reading, though I could be wrong, that it doesn't gel with other parts of your post). The argument is that the Father remains one person because he has separate but not opposite relations to the Spirit and the Son. The Son and the Spirit are two persons because they have opposite relations to the Father. He then argues that distinctions in the Divine Essence can only be distinctions of origin (if they were another kind of distinction God would not remain one Essence), and so the Spirit must proceed from the Son, because will must proceed from intellect.
But to be honest, for my own part I am not sure I understand the distinction between saying the Spirit proceeds "From the Father and the Son" and saying it proceeds "From the Father through the Son". On my initial reading it would seem that both are equally valid, because one might say that will proceeds from body and intellect ('body' used here analogically of course), or one might say that will proceeds from body through intellect, and it seems to me that these capture the truth more or less equally.