Would an account-to-account transfer for online payments be classified as an e-money transaction?

Depends, but I'd go with a No. E-Money in most places (regulators) is classified as stored-value. Are you moving stored value that was issued by you or maintained by you, if yes, then yes, if not issued or stored by you, then No. ?

Account transfer for online payments is just an 'electronic' transfer and would not be considered e-money transactions, unless and until specifically the e-money (stored-value) was being moved.