This userscript will hide your recent actions from public visibility in the
"Who's Online" list by overwriting them with a view to the board index. This does not work 100% of the time, but it should be more than enough to keep your actions generally more private.
This userscript works by hooking the click event on all urls on the forum, and quickly requesting the board index before navigating to the link. Normally you might think it would be better to simply request the board index after each page load, but this does not work since there is a ~8 second cooldown before your last action can be updated again. Hooking only the urls also means that what you are doing could be revealed if you refresh the same page over and over again rather than navigating using the links on the page. As far as I can tell there isn't much of a better way to go about this due to the cooldown on last action updates.
To install this, you will need a userscript manager such as Greasemonkey (for Firefox) or Tampermonkey (for Chrome), or something equivalent since there are multiple others out there.
https://pecon.us/userscripts/incognito.user.js