I have exactly the same poblem since i updated my messenger a week ago.
I am always on invisible before and now it goes green all the time.
It has nothing to do with my internet connection as it did not happen before.
They must have changed something in the new version!!!!
I dont want this,i stay invisble for a reason!

Recently, it appears that if you go to your hotmail inbox, or sign-in any Windows Live services, the web messenger automatically signs you in. As Messenger now allows multiple sign-ins from different locations (search for: Multiple Point of Presence), this will change your Messenger client's status to online.

It seems that if you click sign out on the web messenger, it will never sign you in again. (depending on whether you're using hotmail, or other Windows Live services, the location of web messenger could differ, but I recommend you look for "Messenger" which will have a number of online contacts affixed)