There is an Airworthiness Directive that has been closed by accomplishing its terminating action. But there's a new revision that changes the terminating action. Do I have to comply with the new terminating action to close the AD?
When ADs are revised or new ones issued over the same technical issue as the original, there will normally be language in the compliance requirement to the effect that aircraft that were compliant with the terminating action of the previous AD may be considered compliant with the new one.
If it's not there, such as where a new risk factor was exposed that was not mitigated by the original AD's compliance, you have to treat it as an all new issue. However, where a previous compliance provided a partial mitigation to the new situation, usually there is credit given for that.
There is a fair amount of effort made in crafting these things to minimize the operator burden and avoid unnecessary repetition of work as long as the related risks are dealt with.
ADs aren't 'closed', they are 'complied with'. If the AD has been revised and the new compliance criteria are not met, then you would need to perform whatever measures are required by the revised AD.
There are a number of reasons that the AD may have been revised but in the example that you allude to, the proposed action does not appear to be sufficient to remedy the problem. Therefore a new or revised AD has been issued.
If the revision is mostly a clarification or change in affected aircraft, the identifier usually remains the same e.g. 2018-03-03 R1
If the revision makes a substantial change to the method of compliance, they usually rescind the original AD and issue a new one. e.g.
SUMMARY: We are rescinding an airworthiness directive (AD) for Lycoming Engines model TIO- 540-A series reciprocating engines. The existing AD, AD 71-13-01, was prompted by a report of a failed fuel injector tube assembly. Since we issued AD 71-13-01, we became aware that Lycoming Engines no longer supports Service Bulletin (SB) No. 335A, which was incorporated by reference in AD 71-13-01. The intent of the requirements of that SB is now in Lycoming Engines Mandatory SB No. 342F, which we have incorporated by reference into AD 2008-14-07. The FAA determined, therefore, that this requirement is duplicated by another AD.
I had my A&P comply with a fuel selector AD 2018-02-05 so it was noted as complied with in my logbook. The AD was superseded by AD 2018-07-03 which allows a pilot to comply with the AD, so I made an entry in the logbook to indicate that I had complied with the new AD. This example is the same as the one in your question, though in my case the only additional action required was a logbook entry.
SUMMARY: We are superseding Airworthiness Directive (AD) 2018-02-05 for certain Piper Aircraft, Inc. Models PA-28-140, PA-28-150, PA-28-151, PA-28-160, PA-28-161, PA-28-180, PA- 28-181, PA-28-236, PA-28-201T, PA-28R-180, PA-28R-200, PA-28R-201, PA-28R-201T, PA- 28RT-201, and PA-28RT-201T airplanes. AD 2018-02-05 required inspecting the fuel tank selector cover to verify the left and right fuel tank selector placards are located at the proper positions and replacing those that are improperly located with new placards. This AD addresses the same unsafe condition and requires the same actions as AD 2018-02-05, but changes the inspection of the fuel tank selector cover to a preflight check and allows for various fuel selector clocking configurations. This AD was prompted by our determination to allow the owner/operator (pilot) holding at least a private pilot certificate to perform the preflight check. We are issuing this AD to allow the pilot to do a preflight check of the fuel selector placards.