Facebook – Don’t want people to know I’m online – How to eliminate the green light from showing

No, it isn't possible to know if someone if online and has just set their status as offline. It may arise a privacy concern.

But there's a way to know if someone is online or not by checking their profile, check if they recently liked status, page or anything which can indicate that they are active. But it won't always work.

Update: There's another trick which can tell you if an user is active or not. Setup a group chat on Facebook by sending a message to more than one friend (also add the friends you want to know active status of) and then send a random message, now when someone in the group chat will see the message, there will be a notification like Seen by [name] and if your friends see the message, the notification will be shown for them too.

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.