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 Brazil we officially call it something like "National Enabling Card" (here I'm translating "Carteira Nacional de Habilitação", CNH, in a literal way). By joining the meanings from words "Carteira" and "Habilitação", it takes the meaning of "license". But here's the catch: while the English part of CNH is "Driver License", the original Portuguese name doesn't mention the "motorista" (i.e. the driver). It'd be something like "National License", focusing more on the collective (nation) instead of who is actually being licensed (the driver, the individual, the citizen).
Edit: I noticed that your map is wrong for Brazil. The Brazilian CNH (the newer models) has "Driver License", not "Driving License", among the international languages below the original Portuguese title for the document: English, Spanish and French.