A scimitar is almost exactly the same as a sabre, cutlass and broadsword(not curved - or much along with the backsword).
A falchion type weapon from the same area as a scimitar is called a shamshir.
A shamshir
is a scimitar; it's just a different language (persian versus persian-filtered-through-italian). Saber (or sabre), shamshir, scimitar, talwar, saif... they're all the same thing. They're all essentially saber-shaped weapons: a blade with a radical curve. It would not be inaccurate to put the katana in the same general weapon family, but it is generally seen separately due to having developed completely independently.
A falchion could be seen as going in that group, I suppose. My understanding is that a falchion is used more like an axe than a sword, where scimitars are used more along the lines of a draw-cut, as a slashing rather than hacking weapon. Certainly you COULD hack with a scimitar ("use of a particular weapon" is an area of great argument), but it's really made for cutting rather than chopping.
A cutlass is not necessarily a curved blade, though some were; some were straight-blade short swords. I would tend to just call that a short sword for game terms, since none of them were really long enough to carry the radical curve that defines a full saber. (Definitions get a little loose, obviously, since the smiths weren't exactly trying to make sure they stayed inside certain limits. These are general statements.) A slight curve to the cutting edge doesn't make it a saber.
I... don't even know what you were saying about broadswords. The only actual broadsword I know of is the scottish basket-hilt, which is a straight blade and couldn't even remotely be considered a saber/scimitar. Unless you're referring to an arming sword (or "side sword") which is what D&D calls a longsword, and isn't significantly curved either. (This is another spot of confusion -- a longsword in the real world is a big two-hander, like a bastard sword or claymore...)
Claymore means greatsword and the D&D greatsword is a claymore. The sword on the front of Claymore whiskey is a basket hilted broadsword. (Stuff wiki. A near quote from a sword collection book says "when buying antique swords don't confuse a basket hilted broadsword and a claymore as it displays ignorance." etc, etc and I'm more inclined to believe them.)
Claimh mor just means "long sword", or "great sword" if you're so inclined. But yeah, historically a claymore is a massive two-hander.
Claymore Whiskey is using the later, confused definition of claymore, which comes from British confusion over what the word meant. They started using the term for any scottish sword, which is how it wound up as a term for a basket-hilt broadsword, which is the
other quintessentially scottish blade.
A broadsword looks nothing like a fat longsword.
Quite honestly there's no such thing as a broadsword in real life. Everything that would be called a "broadsword" just falls under "longsword" in D&D terms.
A shortsword is the same thing as a gladius, a short fat tapered stabbing weapon.
A smallsword, closer to what people think a shortsword is, is a shortened broadsword.
I dunno. A gladius or xiphos is more what I think of as a short sword, with a fairly broad blade. A smallsword is more along the lines of a rapier, I would think.
It's kind of confusing because some people say "shortsword" when they mean "arming sword", which is a longsword in D&D terms. Confused yet?