Whenever I turn off chat using the Facebook website, after a while, if I open Facebook again I'll be signed in to chat.
I should explain my setup. I have Windows 8 and a Windows Phone which are both connected to Facebook. In the Windows 8 Messenger app, I set my status to Invisible (the only other option is Available). On my phone I disabled the Use Facebook Chat setting.
After some minutes, when I visit the Facebook website I'll be signed back into chat, and on my PC my status will be Available. All this without me doing anything except for visiting the website.
The problem is I'm developing a Facebook app, and spend some odd hours working on it, so I don't want to be signed in all of the time.
For now I'm going with Joe's answer in this question, but neither of those answers are really solutions to the question (and I don't know how Joe's answer applies to the Messenger app).
How do I sign out of chat and get the settings to stick?