First off, hi everyone as this is my first post here. Been a lurker for some years but this is my first contribution.
Being a Turk myself, I want to give a few firsthand insight.
Now, I've never heard any mention of "Kalij" anywhere, it is known as "Kilij" in West. Turkish word is "Kılıç" and it, as Tibor mentioned, basically means "Sword". But, Kilij, Saif, Shamshir, all basically meaning sword in their respective languages, really came to respond to specific types of swords (well more like sabers) in time, just like "Katana" (which too, basically means sword in Japanese).
Even though Ottoman Kilij, with it's distinctive yelman, is known more with late samples from 17th, 18th or even later centuries, types that were less radically curved yet still including yelman were around much earlier, as it was mentioned here on this forum. Turkish military tradition, as mentioned here in earlier posts, was always mainly consisted of light / medium cavalry, for which curved chopping swords were more useful. And last two pictures showing U.S. Military "Mameluke" sword, while indeed been inspired from Ottoman swords, and there was similar authentic Ottoman swords, is not what is called a "Kilij" today, they are very late military swords, and they had a way less curve then traditional Kilij. Also, they lacked a yelman.
So... I'd say that type "Kilij" as it is known today, do not only respond to late types, and even exclude some late Ottoman swords.
Also, Tibor provided a link of photos from "Harbiye Military Museum", stating that all swords there are labeled "Kilij" even European ones. It is a magnificent museum, I was there two months ago and they really have a very nice collection, also my homecity İstanbul (even though i am not living there for last 14 years) is very beautifu, and I would recommend this museum and İstanbul city to anyone who's looking for a holiday. But... as embarassing as it is to say for me, labels of items there were simply bad. There were many items mislabeled (such as an Italian
Schiavona labeled as a "German Sword" or rapiers labeled as "Swords" etc). I have seen such mistakes even in modern pieces as, a heavy .30 caliber Browning M1919 from WW2 was labeled as "Sub Machine Gun", and a Bergmann MP-18 was labeled as a Sten.
I'd say, during most of Crusades, greater majority of Muslim armies used straight double edged swords. As far as I know, Persians had curved and slender Shamshir before Arab people. But Turkish people used curved swords much earlier.
Also, some Turcoman warriors, who were still pagans, fought alongside crusaders against Arab and Persian peoples, as far as I know. So we might say, sometimes in Crusades, Eastern side used mostly straight swords, and Western side and their allies used straight swords + falchions + perhaps early kilijs :)
I think in time, late curved Shamshir came to symbolize Islamic Crescent, and straight European sword with quillons came to symbolize Christian Cross. And that's the reason they're pictured this way in late artwork.
As a last note, I do not want to sound like a subjective patriot minded person who thinks their cultures [put item here] are superiour to anything else :). I think there is no such thing as "best sword type", and personally I would take European Longsword or Sidesword over anything, anyday :)
Just wanted to add my two cents, cheers.