WhatsApp Messenger zero-day exploit: How to upgrade?
In cases of proprietary software vulnerability coupled with a lack of proof-of-concept and detailed technical information about the exploitation you have to rely on the word of softwares' developers, follow their instructions, and be content that you are safe.
If your WhatsApp shows that you are on v2.19.134, than it must be that. I understand that there are discrepancies in the versions shown to users on Play Store and WhatsApp's website. On Play Store, it shows 2.19.134 as the last version to me as well, and that too updated on May 10, 2019. On APKMirror, there is no stable 2.19.139 as of now, and Appbrain agrees with you here with 2.19.134 as the last update on May 10 it tracked.

Should I just not worry because I'm at the safe version (about which I have now contradictory information).

You do not have contradictory information. Your Play Store shows 2.19.134 and so as the app itself. It would have been contradictory if these two mismatched.

Should I just wait until WhatsApp Messenger updates itself (which presumably will happen soon, or maybe can be explicitly triggered).
Should I install the version at https://www.whatsapp.com/android/ instead of using the Play Store (I'm not even sure that would work).

The Facebook advisory clearly mentions that versions prior to 2.19.134 are affected. As long as you have 2.19.134 or above you are safe (according to Facebook). There is no need worrying about why the official website has some higher version available. You can always try installing that one. If the signature mismatches with the sideloaded version the app won't install, and you should put this matter to rest. If it matches you would have the latest app and that should definitely work.