Is a CFI's 1st-time pass-rate based on the student or the instructor's first attempt?
I believe that your pass rate is based upon your number of students who passed on their first attempt after endorsement vs the number of students you endorsed for the checkride.  I don’t believe the FAA makes any other distinctions here, at least not for §61.197 requirements for the renewal of a flight instructor certificate nor for 89001, Section 13, Issuance of a Gold Seal Fight Instructor Certificate.