And finally, the last getting in depth with the four guards.
Since there is not much variation to be had, I threw in a fifth (!?) bonus guard.
Fourth guard according to Ringeck/Danzig: "Hold your sword in your right hand and set the pommel on your saddlebow, with the point facing him." Clearly I should have used my medieval saddle for this demonstration 😂
Anyway, the only reason I can think of that would distinguish the forth guard from the third, is to use the arcon-plate of the saddle to make powerful thrusts against an armoured opponent, using it basically like a lance secured to a lance-rest.
Fifth (bonus-) guard.
Ringeck/Danzig: "Hold your sword with the right hand by the grip and hold the middle of the blade with the left hand and hold it against your saddle." This is clearly a technique for harnishfechten all though Talhoffer has some unarmoured mounted techniques involving wrestling, the halfsword techniques, generally go hand in hand with harnischfechten.
From here you can defend powerful attacks and make precion-stabbing in gaps of your opponent's armour, stiffening the blade with your left hand and of course: wrestling!
Unless you are proficient with riding 'on the belt' I'd say halfswording on horseback is likely intended for wrestling possibly ending with both parties on the ground.
On a sidenote you'll notice I'm using a longsword for this 'demonstration', partly because it is depicted in the "Goliath" manuscript and partly because it's easier to see the positions with a larger sword. I would however pick an armingsword any day rather than a longsword for a mounted duel. Even with a fairly good blade such as the Albion Lichtenauer, the balance is just all over the place when used one-handed, that is atleast my oppinion on the matter.