In Korea, we have many companies providing drivers who actually go to a drunken man's site and drive him or her home, getting paid in cash at the end. I don't think English speaking countries have this service. If you did, what would you call this service among the three below? Or something else?
ex)temporary driver or designated driver or chauffeur service?
We actually do have these services nowadays. I don't think there is a generally accepted term for them. It is definitely not designated driver. A designated driver is one of a group of friends who volunteers to abstain from alcohol in order to drive his or her friends home legally.
not a teacher
Companies in Australia which offer this service often advertise themselves as a "designated driver" service, probably because the term is well established and so the specific nature of the service is understood. That is, they go to meet the client and drive them home in the client's vehicle, but they are not a general chauffeuring or taxi company.