A cubit is a general form of measure - it is approximately the length of your arm, from the elbow to your finger. As such, a cubit isn't specific to a culture, it is specific to a man. On average, 18 inches is a good approximation, but that can differ from person to person - for some people it might be 15" for others, 21". The various cubits that you see reflect later legislative (and historians') attempts to "standardize" the measure. The exactitude you look for in modern groups didn't exist then.
As Robert Creese points out in his wonderful book World in the Balance: The Historic Quest for an Absolute System of Measurement, human based measures were never intended to match the precision we look for today. Cecil Adams points out that the 'mile' has varied in length over time as well. "18 inches" is specific. Cubits are not that specific.
As such, there are two good options to answer your question:
- God was saying to Noah - "Use your cubit."
- God didn't care about the exact dimensions to the precision you are looking for.
And of course Narnian is right on when he points out that Persia, Rome, Sumer, and Egypt were hundreds of years after Noah.
Incidentally, Creese specifically mentions that there are even different kinds of cubits - sacred cubits were 25 inches!
Indeed, Taylor said, the pyramid had been built to provide “the measure of the Earth” for humans. Furthermore, the unit used to measure the stones was the “sacred cubit” (about 25 inches) that had been used by God’s chosen people throughout the ancient world in their holy construction projects—Noah in his ark, Abraham in the tabernacle, and Solomon in the temple.7
Crease, Robert P. (2011-10-17). World in the Balance: The Historic Quest for an Absolute System of Measurement (Kindle Locations 2295-2298). Norton. Kindle Edition.