It can be a combination of all three of them, a lot times they are in fact nicknames for people vaguely from the time period. One of the better known ones being Charlemagne (Charles the Great), which crops up pretty rapidly after his death in 814, and many sources refer to him as 'Carolus Magnus', and from there it eventually morphed into Charlemagne. However it can also be something historians tack on, Louis the German would be a decent example of this. In the Annales Bertiniani they usually refer to Louis as something like 'Louis, king of the Germans', and then in the 19th and 20th centuries German nationalist historians have tended to incorporate him into a progression of German history, so Louis the German is now the accepted name for him. Others have been contested, for instance Charles III or 'the Fat', whose moniker is not from his own time period but a few centuries later and recently Simon MacLean has argued against calling him that in his book Kingship and Politics in the Late Ninth Century. So in summary, a lot of times they do come from the time period but they can also come afterwards, or be something historians come up with (or follow along with). In the Carolingian period this happens quite a bit to distinguish the numerous Pippins, Charles's, and Louis's. Hope this was helpful!