AFAIK, despite the numerous things that you can now submit online, a COFP is still a paper document that is served by post. Payment can be made online, but the paper form would have instructed you to provide your licence details separately by post.
What the statute (
section 75 of The Road Traffic Offenders Act 1988) requires is that payment is made to the "appropriate person", and that either your name, date of birth and licence number is provided to the "appropriate person", or that the "appropriate person" is satisfied as to your identity.
The appropriate person, for the purposes of payment is the fixed penalty clerk.
The appropriate person, for the purposes of identification is also the fixed penalty clerk.
If the requirements are satisfied, the driver cannot then [lawfully] be prosecuted for the offence to which the COFP relates (subject to not being liable to tot up).
As UK driving licences incorporate the holder's date of birth (in the form YMMDDY), it is not immediately obvious that satisying the fixed penalty clerk of your identity when making payment, and/or providing your date of birth by providing your licence number, does not satisfy the requirements of the legislation to avoid liability to prosecution for the offence, regardless of whether the police decided to make up and apply their own rules.