How to use different domains for multilingual sites and keep users logged in?
A cookie can only be read from the same domain that set it. This is a fundamental security feature in browsers to prevent other sites from potentially stealing login sessions, and there is nothing simple you can do about it.

With that said, the comment in the settings.php file seems to contradict my statement, and I'm honestly not really sure how to intrepret that. Perhaps it's only meant to be relevant to sub-domains, which are allowed to share cookies, but if so the comment should probably be re-phrased.

There are a number of ways to work around this, where the easiest is to use language prefix sub-domains instead of separate domains.

Baring that, I believe the "best" one is to use a Single sign-on system. There's a comparision of different SSO modules here. This could be a Drupal based solution, which is the way that is most likely to "nicely" integrate with your current Drupal sites, but that's not really necessary.

The Bakery module which is used by *.drupal.org itself is relevant to mention, as well as this related DA answer How drupal.org single sign on works

More info about cross-domain cookies in a Stack overflow question.