The FAA requires...
The FAA requires that any instruction be recorded....as far as I know, 'how' this is done is not specified (i,e, in the instructors logbook, company training records, etc.). But, it is required that al this be recorded...each session, contents of each session, time for this, any endorsements, recommendations for checks....everything.
Having this, it would be logical for the instructor to record this. And, it's important to have detailed records....again, however they're kept....so, when the illustrious student makes a booboo, and the FAA comes to you, the instructor is asked questions, you have some documentation to answer those questions. If there's an accident, you can bet the FAA will ask a lot of detailed questions.
Fly safe,
PantLoad