Based on comments from Batman Incorporated writer Grant Morrison that the sword in question belonged to Batman's oldest known descendant, we assumed that the sword was Gaweyne's, and that the blade pictured at right was the same as the one pictured at left, with minor variations in the look accounted for simply by different artists and different creative eras in Batman.
It was a bit of a trek to finding that information, given that Morrison had cited Batman: The Shadow of the Bat (a series that Grant wrote for years) as the source of the reference, and so coming across the one-shot was almost incidental in our research. That, combined with the differences in the swords, led us to feel pretty comfortable saying the blade came from The Scottish Connection but less confident that we had it exactly right, and we reached out to artist Chris Burnham via Twitter for confirmation.
"Kinda sorta. The sword seen in the Scottish Connection isn't Gaweyne's sword. The bad guy finds it in a treasure vault," Burnham told us. "Gaweyne was a Knight Templar, so the armor and sword look like typical Templar material, with bat-affectations for fun."
Which sounds like confirmation that it's Gaweyne's sword from The Scottish Connection that killed Damian, but is a clear rejection of the idea that the swords pictured above are the same one. Just as well--it probably would have been difficult for Bruce to get a bloody, antique sword through airport customs.
As for our looks at the suit of armor in Wayne Manor, Burnham was a bit more at a loss, saying, "I haven't actually read Shadow of the Bat #45, so I don't know if that connects directly to The Scottish Connection."
Batman: Shadow of the Bat #45 featured another of Bruce Wayne's descendants, an abolitionist who fought slave traders with a cutlass. The armor was featured briefly, as police walked past it in Wayne Manor on the way to the discovery of Joshua T. Wayne's skeletal remains in the wine cellar, and was not a plot point, nor was it seen particularly clearly.
When we followed up, he later added that it was probably but not definitely the same armor seen in the Batcave during Shadow of the Bat #31 (which we assume to be the same one from #45, since how many suits of armor does one really need hanging about the house?), saying, "I guess it's basically a different artist interpretation of the armor from Zero Hour (although [to be honest] I never knew about it)."