On the EASA logbook,
FCL.050 Recording of flight time
The pilot shall keep a reliable record of the details of all flights flown in a form and manner established
by the competent authority.
In the UK that is Article 228 of the UK ANO where the only reference to differences training is in relation to National licences.
FCL710 states:
(c) The differences training shall be entered in the pilot’s logbook or equivalent record and signed by
the instructor as appropriate.
What they write or how they sign it is entirely up to the instructor. In the case of Electronic Log Books an equivalent record may be required.