In English, the name of document that entitles someone to drive a vehicle differs, with around generally 6 permutations. Driving/Driver/Driver’s and licence/license.
As a noun, “licence” is generally how you would spell the verb using British English, whereas in American English the noun is spelled the same as the verb; “license”.
Driving vs. “Driver’s” is more subjective in my opinion. It is an authorisation for the act of driving, so it being a “Driving” licence/license is logical. As the same time, the document is in the possession of the driver, so “Driver’s” is also equally as valid. A handful of countries use “Drivers”, which is just sloppy, as it doesn’t make any grammatical sense.
I tried my best to compile data on all countries which mention the document in English. In Australia, Canada and the US, licences are issued by state/territory, so I’ve included their differences.
I only included countries for which an English version of the name is on the actual licence. On many EU licences, the English is written very faintly on backgrounds. For many smaller countries I couldn’t find examples of the document. In South America, ‘Licencia de conducir” was most common, but a few permutations in Spanish. On the African continent, the French “Permis de conduire” was also fairly common. Multi-language licences with English, French and other languages was also common. I only picked out the English translation for this map.
In my subjective opinion, "Driving licence" feels most right; but as this map illustrates, it’s a diverse interpretation. For licence/license, the difference between C/S is almost indistinguishable in a small font and in spoken word. Some evident US/UK influence on the map.
https://old.reddit.com/r/MapPorn/comments/1fvftxw/drivingdrivers_licencelicense_oc/
In North Carolina in the US, we might have official documents calling it a Driver License, but having lived here almost my entire life, I would use "driver's license" in casual conversation.