Can facebook friends see that your online when your chat is offline?

[E]

The same happens to me as well with the recent introduction of FB chat client update. My chat is set to offline but when I first login to Facebook , for a very short amount of time I appear online on my friends lists and then switch back to offline. We tested this weird bug after we put two laptops next to each other and logged in with different accounts... If some of your friends happen to send you a message right at that moment, when Facebook makes you online, then you receive a message as if you are online...

As a computer scientist, I believe the reason for this bug is that Facebook set the default chat option to online so that for a very short amount of time until Facebook loads your privacy settings and realizes that your chat settings are offline, it allows others to see you as online...

Personally I find it very annoying but I couldn't find any solutions yet.