Hungarian kocsi referred to a wagon from Kocs. German and French carried the vehicle name toward English. The later tutor sense used the idea of carrying a learner through an exam.
The display selects the shared borrowing route and does not score the period label of earlier English coche. Sources give different German spellings; these are not separate chronological steps.
Source
Merriam-Webster: coachWord History supplies the borrowing route and Hungarian place-name connection.