Questions about the Thrusting Broadsword, Backsword and Cavalry Saber

Hi folks.

I ignore if this question was already put, but my researches on the forum gave nothing.

My group of adventurers stemming from a TL3 arrives in a region where the military technology is TL4 (without the black powder).
In this region most of men carry Backswords or Cavalry Sabers.
I take in count than this swords are avalaible in more quantity than there ancestors.

These weapons are very similar, nevertheless for somebody who has a Thrusting Broadsword of the 13th century what could attract him to take a Backsword or a Cavalry Saber?

Of course, I take into account the problem of familiarity between weapons (-2 to skill) and the protection which offers a complete handguard to the hand of sword. But the thrusting strike of these weapons make fewer damages than a thrusting broadsword (Th+1 vs Th+2).
Moreover, I don't understand why there are a damage difference between a Thrusting Broadsword and a Backsword if they are the same blade?

And form the description of LT p.56 Cavalry Saber: A curved sword optimized
for one-handed use from horseback.
It modifies anything, or gives any advantage?

Thanks in advance for your suggestions.

Questions about the Thrusting Broadsword, Backsword and Cavalry Saber