The word 'Relevant' in this sentence relates the the Examiners authorisation to type. The examiner was relevant to both SEP and TMG.
No it does not. You can only exercise the privileges of a Class rating in an Aeroplane of that Class! This is fundamental to all Class and Type Ratings.
I quite agree the CAA should know, but fundamentally the Examiner is a Pilot and the man on the phone is a Clerk so really its down to the Examiner to know his privileges, after all he will be examined on them when he renews his Examiner authorisation.
(a) Privileges. Subject to JAR–FCL 1.215(b) and (c) above, the privileges of the holder of a type or class rating are to act as a pilot on the type or class of aeroplane specified in the rating