Bonus: my friend tells me, "The notary is likely someone who works in an adjacent office and not the registrar herself. She literally cannot sign as you've written it--California law prohibits it. The registrar is the document custodian, and all the notary does is certify that the document custodian is who he/she says they are. A notarization is solely a confirmation of identity (in this case the registrar's)."
Update: I received a notarized copy of my diploma. Basically, it's a copy of your diploma with a piece of paper attached to it signed by the registrar indicating as follows: "I am the University Registrar at UNIVERSITY NAME. I hereby verify that the attached diploma is a copy of the original." A notary reviews the diploma copy to make sure it's an accurate copy of the original. The registrar recites a short statement and the notary stamps a piece of paper and makes a notation in her book. That's it.
The registrar--a delightful, smart woman--said she had nothing to do with the law school application process, so she wouldn't certify or write anything other than what her databases showed. She referred me to the law school's assistant dean of student services for the letter I requested. Luckily, I knew the assistant dean from my time at the law school, and she is an amazing person. I got the letter.
The registrar told me national databases do in fact exist that collect student data. See HERE (NSLDS) and HERE (National Student Loan Clearinghouse). However, from my research, such databases are not publicly accessible like the State Bar's website, leading to this ridiculous business of notarizing diplomas and transcripts, which transfers money and time from regular people to institutions and their employees.
Why my law school diploma had to be verified by the undergraduate registrar rather than the graduate institution itself, I don't know. For a smaller private school, I suppose it saves overhead to consolidate graduate program information into a single central database that includes undergraduates. I lament once again the American predilection not to be consumer-facing in terms of saving time from the perspective of the consumer, but to organize affairs in order to save costs from the perspective of the entity.
A final note: apparently, some law schools, during their application processes, don't actually check any database to see if an applicant graduated from the listed undergraduate institution. They just request a verified official transcript.