The quiver is an interesting question. After a lot of study it seems to me that two main types of quiver were used int he East. A „flat” type, related to the Mamluks, Turks, Crimean Tatars; and the „hourglass” form related to steppe nomads. It is proven by period illustrations that both type were known parallely and I think a lot about why would a warrior choose this or that. I wrote an essay on my Hungarian site about this topic, here you can see it and the pictures proving that both type existed in the same place and time: nomadok.gportal.hu/gindex.php?pg=32489510
David Nicolle wrote in one of the Osprey books that the hourglass-box type was used by real steppe nomads living on horseback and they needed to protect the arrows from weather. The flat type was used by professional soldiers at the perimeters of the steppes who didn’t need such protection from the weather, but their rate of fire was higher because the quiver was more opened.
I argue with this theory because steppe-born Mongols also used flat quivers and Mamluks out of the steppe region used hourglass type, too, as Nihayat al-su’l proved it. In my opinion (as I have used both types on foot and on horseback, too) the hourglass type is comfortable only on horseback, it is difficult to walk or run with it. But on horseback it is perfect, it’s out of the way of other weapons, hangs under the belt while the arrows can easily be drawn. In the flat type the arrows can easily be destroyed by other (even own) weapons, but it is much more comfortable on foot.
I also argue that the loading is quicker from the flat one. In the hourglass type the arrows are upwards with their heads (as Nihayat al-su’l proves it, too, besides in situ archeological findings, and tons of period illustrations) and loading from it can be as (or more) quick as from the other if we know the method.