>> "I'm wondering whether the details here are old or whether I'm doing something wrong? The trial extension web page doesn't seem to provide an offline/online option."
Offline trial extensions are deprecated and only available for unverified trials. For verified trials, use either the regular trial extension, or use the offline verified trial request: https://wyday.com/limelm/help/trials/#ut-veri-req
>> "Either way, should I assume that extending a trial offline should involve a trial extension key along with UseTrialVerifiedRequest() "
No key is needed.