England operates using the 2013 edition of Approved Document P. This reduced the number of special locations and introduced the concept of Third Party Certification. The process goes like this... you engage the services of a spark, spark designs the installation, you install it and they inspect it at key stages they specify, then they inspect and test it, issue an EIC and a building control notification I believe.
Wales operates using the 2010 edition of Approved Document P, which has a much bigger list of special locations and no Third Party Certification.
In either location you can issue EICs, MEIWCs and carry out EICRs providing you are competent to do so. You can notify the council of your plans ahead of time pay them a shed ton of cash to inspect and sign off, show them your quals and hope they deem you competent and then notify them ahead of time but just send them the EIC and have them sign off or be a member of a scheme, do the job and notify yourself after the event.
So yes, your statement 'Notification cannot be done by a third party' is correct.