Mostly correct.
Just fly a LPC SPA with a UK/FE(PPL); as soon as you have completed the flight satisfactorily he/she can renew your Single Engine Piston Class Rating and you send a form off to the CAA for their info. As soon as the UK/FE(PPL) has signed the Certificate of Revalidation in your licence, you may fly with JAR/FCL PPL privileges.
Normally you would fly a 1 hr dual training flight as part of the 'revalidation by experience' requirements in the second 2 year period to revalidate your SEP Class Rating; however, an FI cannot 'revalidate the Rating', i.e. sign the Certificate of Revalidation in you licence - that must be done by an Examiner.