The linked documentation should work fine as long as your user update code is calling the onAfterUserUpdate trigger as that's what CB Profile Update Logger is using to log updates. Is your custom implementation happening in frontend or backend? Frontend and backend have different triggers and CB Profile Update Logger has to be explicitly configured to log backend changes.
It might just be a plugin loading issue though. Be sure you've also the below code.
Code:
global $_PLUGINS;
$_PLUGINS->loadPluginGroup( 'user' );
See the below documentation regarding loading API externally.
www.joomlapolis.com/documentation/127-community-builder/279-tutorials/18357-including-cb-api-for-usage-outside-of-cb