Tony Verhulst wrote:

In the U.S. instructors only need to maintain a record of test
recommendations, solo endorsements, and pre-solo written tests. During
my initial instructor exam the inspector asked me "are you required to
log each lesson given in your log book"? My answer was "No, but I think
it's a pretty good idea". The inspector responded with a vigorous nod of
the head.

See FAR 61.189

The CFI isn't required to log it in the CFI logbook,
but the CFI is required to log it in the student's logbook.

(a) A flight instructor must sign the logbook of each person to whom
that instructor has given flight training or ground training.

I have a bunch of white labels (stickers) that I use, and
write out the entry with the pilot's name, and give it to them
to put in their logbook (if they didn't bring a logbook with
them). For brand new pilots, I give them a new logbook
(about $3 each as I get them in bulk).
--

------------+
Mark J. Boyd