It's good that you have started reading LASORS because that's the definitive source of information. (Well, almost, since LASORS also contains some good advice that's nevertheless not based on any legal text.) But you should not read it selectively, but the whole relevant section.
The bit that you missed is at the start of para F1.4 (page F10), which says:
A SEP (Land) and/or TMG class rating can be revalidated
by flying experience by producing logbook evidence to
an appropriately authorised JAR-FCL Examiner, before
the rating expiry date has passed, of the following flying
experience completed within the 12 months preceding the
rating expiry date.
And then follows the text you refer to (page F11) about the 12 hours flight time, 6 PIC hours, one hour with an FI and so forth.
So you need to present the evidence of your experience, including the signature of the FI in your logbook, to an *examiner*, who will then review the paperwork, sign the relevant forms and shake your hand to congratulate you. This can be sorted in less than five minutes and is usually done for free, or for a very modest fee.
At my club we typically leave all relevant stuff (original logbook, original license, paperwork, fee) at the front desk and the resident examiner will go through it all when he's around. You then pick up everything, including the signed paperwork, when convenient some days later. Saves you from having to set up an appointment.