Is one spelling of Mahomet or Muhammed etc. correct?

I wonder why there are so many spellings of Mahomet or Muhammed or whatever he is called, whether any spelling is  'correct' and if so which. 

I came across this on the net:

Inexact spellings such as "Mecca" and "Koran" entered the English language a long time ago and have become so entrenched that they are now difficult to eradicate. In old books the Prophet's name is frequently spelled as "Mahomet" and this is still used to some extent today. There is no logical reason for it because Muhammad is one Arabic name that can easily be rendered in a way that is both phonetically accurate and faithful to its written form. 

Assuming this is true, it seems to me when writing English the correct spelling is surely Mahomet, if this entered the English language a long time ago. Incidentally the Latin version is Mahometus, which reinforces this view. What Arabs say is beside the point. Romanians call Jesus Christ Isus Hristos without caring what the Aramaic version is. (I think His Blessed Mother called Him Eashoa or thereabouts. Joshua is much closer to that than Jesus).

So that's sorted out.

However I think correctness is a very fatuous ignis fatuus and we should all spell foreign names as we choose.

There is a town called Mahomet, Illinois, about which Wikipedia curiously says

Although the unusual spelling, which is French for Mohammed, sometimes confuses newcomers, the pronunciation of Mahomet is muh-HOMM-it.