To aid in clarification of the term "student", read the first regulation under Subpart C - Student Pilots
61.81 Applicability
This subpart prescribes the requirements for the issuance of student pilot certificates, the conditions under which those certificates are necessary, and the general operating rules and limitations for the holders of those certificates.
Ergo, the FAA defines a "Student" as one who holds a Student Pilot Certificate.
Once you hold a Private Pilot Certificate, these rules in Subpart C do not apply to you.
61.31(d) applies to other than student pilot certificate holders. And for 61.31(d)(3), the only endorsement required is a solo endorsement - the pre-solo knowledge test is not an endorsement.
Having said all that, I agree that a knowledge test is a good idea and should be accomplished. But is is not a legal requirement.