Please show/tell people where you log out instead of just pointing to the FAQ. I have been a member for quite some time and I had NO idea where to log out. I had to go to meta.SO and do a search to find How do I log out of Stack Overflow?
I am very open to suggestions for rephrasing that message (also, improving the login process for chat.SE is on the radar), but what you're suggesting is not going to work.
First of all, there's no "requirement". The point of the "possibly logging out before" part (which was added only a few months ago) and the reason for adding it is the following:
When people see
they often turn this into
and answer "But I am already logged in to foobar.stackexchange.com!"
What we're trying to convey with the "possibly logging out before" is that this answer doesn't count. The "Your login data seems too old" error message is a very generic one; it just means that there is some sort of global login data stored in your browser, but it's invalid in some way. Exactly how it is invalid is something we cannot tell you, because that would be a huge security hole.
So what we tell you instead is to get a fresh cut of global login credentials, and you get those at the time you log in to a site. That's why we ask you to log in somewhere, and that's why "logging in" and "being logged in" is crucially different here. And if as "somewhere", you choose a site where you're already logged in – well, you obviously can't log in again until you log out before.
So much for explanation; if you have good ideas how to phrase this in a short and sweet message: Hit me. Using a different language that has less ambiguity between "logging in" and "being logged in" doesn't count :)
Now for your request. You're asking us to tell you where to log out; the problem is that if we knew where you're logged in (if you are at all), we wouldn't be having this issue in the first place. But as explained above, the "logging out" part isn't the main point anyway.
If, on the other hand, your main point is about the discoverability of the logout link on the SE sites (it's not quite clear from your question), then this really has nothing to do with chat. In this case, maybe open a new question and explain why you didn't find it and where you would have expected it instead.