If the note doesn't say "applied first to all unpaid interest accrued through the due date" then why would it be applied that way? "All accrued unpaid interest" seems (to me) fairly straightforward - all accrued interest through the date of payment. If that's after the due date, then more interest has accrued since the due date, and should be paid. If that's before the due date, then the additional interest through the due date hasn't accrued yet.
Nobody's perfect, not even a perfect stranger.