I don't see a problem with the Principal or VP having access - they are in charge of the school, they should know the details that go on with each kid including those details from the clinic. Registrar - maybe shouldn't have access to your notes but in my opinion should be able to pull up your logs to see who you sent home ill so the attendance records can match - this comes into play if there are truancy issues with students.
I doubt the principal or VP even look at the records - I know mine don't, if they have questions they usually come straight to me. My Registrar doesn't look either - easier to pick up the phone and call me regarding clarification of a student going home ill - and with that I don't give details - just the general reason - vomit, fever, injury etc so she can make a note of it on her end incase truancy issues come up.
Even if your admin doesn't consider the clinic documentation HIPAA compliant it still needs to be FERPA complaint - the health records are part of the education records - there are privacy laws regarding education records.
I wouldn't worry to much about it nor would I document another way - if your district set up online documentation I would stick with that.