The requirement is that providers can evidence that the names and date of birth returned match official identification. A record that this has been checked can be kept in whatever way the provider chooses.
A signature on a form or a checkbox in a student record system would both be fine as they would allow verification that the checks have taken place.
Whichever way providers choose to record that the necessary checks have been done, it’s worth ensuring that there is a way for the staff member conducting the check to add a note.
For example, a note to explain the difference would be useful if the student has used a shortened version of their name on their enrolment form but their full name is recorded in the student record system and the ILR.
This does not need a complicated process. If the checks are recorded on enrolment forms the staff member could just add the full name onto that form.