Yes, to clarify for the OP.
The self-'certification' in the self-certification scheme relates to the Certificate of Compliance which they issue to the customer when you tell them you have done notifiable work. It (the self-certification) has nothing to do with Electrical Certificates which should be issued for all work and can be completed by anyone.
By the way, the self-certification schemes do not see your certificates when they issue their certificates.
They only view your certificates when assessing whether you can continue to self-certify.
If you choose not to register with a self-certification scheme then when you wish to do notifiable work you must inform and pay the LABC. On completion of the work the LA will issues a Certificate of Completion to the customer.
So, you see there are different certificates, some of which - electrical certificates - can be issued by anyone, but depending on whether you are registered to self-certify or not someone else will have to issue a certificate.
With non-notifiable work you still have to issue a certificate but that is the only certificate; no other certificate has to be issued.
Clear?