Google Hangouts – How to Appear Offline

Based on comments I know that you don't have any other devices connected (which is a big first thing to check if others have this problem in the future). And you have tried the 'traditional' turn off chat:

This is not a duplicate of How can I log out of Facebook chat?. It says “Turn off chat” on that question's answer but I say “I turned off chat but it still show me as online” — these are different issues.

Based on other reports within facebook (e.g. I am seen online even when I am offline) it seems like many others are having this problem and that it is, indeed, a bug. However, a temporary workaround that can be successfully used is to manipulate the Turn on chat option for only some friends mode.

To turn chat on for just some friends

1. Click gear in the chat sidebar
2. Select Advanced Settings
3. Click Turn on chat for only some friends
4. List the friends you want to turn on chat for

If you select just one friend to appear online for you should successfully appear offline for others.

As I mentioned in comments, here is the link through which you can report a bug with facebook chat.

Based on this answer in Google Support, the "last seen" time indicates the last time someone was "seen" by Google Hangouts. They don't necessarily need to have been talking to someone at the time. (For instance, I'm not in a Hangout or otherwise chatting with anyone, but I've got a green circle next to my name for my contacts because I am logged in to Hangouts.)

If they're not currently available in Hangouts, they are either off-line, idle, or have turned off the "last-seen" setting in their profile.