I use the term "basket-hilted" to refer to the type of sword rather than the type of hilt. I understand that basket-style hilts may be found on any number of swords and daggers, but the term, as far as I understand, makes specific reference to the broadsword, which is what I intended. The term can also refer to the basket-hilted backsword, which I would classify, along with all backswords, as a scimitar. Thus, you find a distinction among scholars between basket-hilted broadswords and basket-hilted backswords. The argument I was making, to which you replied, was that the basket-hilted broadsword, which I suppose we could just call a broadsword, being a one-handed weapon, is lighter and more gracile (which doesn't necessarily mean the same thing as graceful) than the longsword, and so could justifiably fill the niche presently inhabited by the rapier (1d8 finesse, not light). Given what you presented (which I quote below), I think you agree with me.
I differ in that I see the arming sword/knightly sword as the equivalent of the short sword. It's the basic, one-handed sword. Notice that is already has finesse, so I'm not making the argument that the broadsword is significantly more gracile than the arming sword.
That seems like a good solution to the OP's problem.
I would be fine with an arming/knightly sword being a short sword. I put it in the next category based on earlier editions of D&D defining a short sword as having a blade up to 24" in length.
Agreed on the broadsword, and I only called out the 'basket hilt' because the type of hilt doesn't have anything to do with the nature of the sword itself, and I didn't want people to latch onto that terminology. And yes, filling the 1d8 range with the broadsword makes sense.