What adjustments do you have to make if partner leads "third highest" instead of "fourth highest" to open?"
Yes, The Rule of 12 is used, except it is the Rule of 10 and 12. Third best is led only from a holding 3 or 4 cards and fifth best is led from holdings of 5 or 6 cards. Opening Leader's partner is expected to apply the result of subtracting from both 10 and 12, with the information available from seeing Dummy and hearing the bidding, to make an appropriate assessment on Opening leader's holding.
The rationale is:

Remove the ambiguity from leads of 3rd best from 3 and 4th best from 4;

More readily allow partner to identify holdings of 5 cards by Opening Leader; and

Combine more consistently from Coded 9's and 10's Leads (vs NT), where Opening Lead of Jack denies the Queen, and that of either the Ten or Nine is either top of a sequence or show the next higher card in a broken sequence, as any of: AJT, KJT, AT9, KT9, QT9.