"Other notaries do it"
From previous discussions, I recognize your quote as being from the California handbook. I'm from Vermont, so I've never taken a notary class (but will take one this month now that the SOS is finally putting some on). I would agree with your interpretation of "OR other circumstances". For example, California no longer allows personal knowledge to establish a person's ID. But suppose you happened to be behind the signer in line at the bank yesterday, and happened to hear the signer use the name Charles Arthur Floyd. Today the signer wants to use the name John Doe. That would be an "other circumstance".
Notice also that your handbook says "Paper Identification Documents – Identity of the signer can be established by the notary public’s reasonable reliance on the presentation of any one of the following documents..." The handbook does not define reasonable reliance. I've seen in forum discussions that when people called the SOS's office, they got different advice about this on different days. Some got the advice that you can't use a combination of ID documents to link from the name on the ID to the name on the document. If I were in California, I would not use a combination of documents to try to link between substantially different names. But I would recognize there are common variations in name use, and that forms and ID card issuers often put limitations on what parts of their name people are allowed to use. An ID card issuer may refuse to include the full middle name, the middle initial, a double surname of a Latino, or a suffix like Jr. Traditionally in America, it is the first and last name that are considered important. If the first and last name on the ID match the first and last name the signer wants to sign, I think "reasonable reliance" would allow the notary to look at a additional evidence to add a middle name, additional surname for a Latino, or suffix like Jr.