Hi @Eden-McGorlick and @Anton-2, thanks for bringing it up! It is an excellent question, and I will give you two answers: an easy, and a full one.
- Easy answer.
The required answer in this card is an expression: вести машину (автобус / пр.)
See the choice of words for a description of silent gestures by RusCorpora - a national linguistic corpus:
It is also possible to say “водить машину”. When we say "вести машину" we talk about the action in the moment of driving / a single action. When we say "водить машину" we talk about a more abstract skill of being able to drive, driving multidirectionally and iteratively.
- A full answer.
This matter is better explained turning to the ancestors of the modern Russian: Proto-Slavic and Old Church Slavonic (that is way, way back, even to the times before language had a writing system), and here’s why:
Actually, we are not just dealing with two verbs: a more concrete вести & and a more abstract водить. We must look at at least 4 verbs to get the gist of things: вести / водить; везти / возить.
At first, let’s look at the definitions of each word in the modern Russian:
- вести (from Proto-Slavic *vedti or *vesti, Old Church Slavonic вести), currently has about 11 definitions, mostly described as “to lead / to conduct” (e.g. вести собрание). Due to an abundance of definitions also has a few common expressions, one of which is “вести машину”, to drive.
водить (from Proto-Slavic *vedti or *vesti / *voditi, Old Church Slavonic вести / водити), currently has 3 definitions, and a few common expressions. Mainly describes “facilitating a movement of somebody / something”, plus the same action(s) as the verb вести, but more abstractly / iteratively / not in one go (Водить детей на прогулку, водить автомобиль).
везти (from Proto-Slavic *vezti, Old Church Slavonic вести), currently, in its first sense, has 2 definitions, meaning “to carry somebody or something by vehicle” or to deliver (везти детей на автобусе в школу, везти самолетом).
возить (most likely from Proto-Slavic *vezti, Old Church Slavonic вести / возити), currently has 3 definitions, mostly meaning to transport/carry something/somebody by vehicle multidirectionally and iteratively, as opposed to the verb везти (автобус возит нас каждый день).
An attentive learner already noticed that there are two big relatives to all these words: a Proto-Slavic *vesti or *vedti and a Proto-Slavic *vezti. Proto-Slavic language is a reconstructed proto-language of all Slavic languages, so it doesn’t have a writing system, which is why we spell these words with an asterisk.
The Proto-Slavic verb *vedti or *vesti seems to carry the meaning of “leading, conducting”. Notice that the verbs with this anestor possess “leadership qualities”; one needs to “know something” in order to be able to conduct an activity: вести собрание, водить детей на прогулку, водить автомобиль, вести машину.
Reconstruction Proto-Slavic/vesti: https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/vesti
The Proto-Slavic verb *vezti seems to carry the mearning of “transporting something”. Notice that the verbs with this ancestor reflect either facilitating a transportation of smth/smb, or literally carrying smth/smb by a vehicle: везти детей в школу на автобусе, автобус возит нас каждый день.
Reconstruction Proto-Slavic/vezti: https://en.wiktionary.org/wiki/Reconstruction:Proto-Slavic/vezti
In Old Church Slavonic the stem вести starts being associated with both meanings: to lead, and to transport, which, probably eventually brings us to our expression вести машину. Also it might be resonating with the Russian speakers so well because one needs to have certain knowledge in order to drive (read: lead / conduct a car).
@Anton-2 thanks for being on the moderator watch and flagging it up with us! This one, indeed, was a big question that required some linguistic heavy-lifting. I have flagged this card up with our linguist to think of a note on the card that would retell everything that I've mentioned above in a couple of words.