Well, it actually DOES work on browser. I tested it with my friend.
The only change that I noticed on browser when I've used "Invisible" in Miranda is that it doesn't show green spots (for online status) anymore as long as you don't click to chat with somebody in browser. After you click on him, contacts gets their green spots notification for Online status. But during all that time, you're online for them (either with or without green spots on their names).
If I remember well, before the problem arrived, green spots would show up automatically each time you click on chat for all the contacts who are online.
So, now only 2 statuses work in Miranda (Online and Offline), same as in browser. Invisible started to act like Online. If I'm using "Invisible" in Miranda, I'll see that my chat is off in browser, but that still shows me as online to other contacts. After I click on chat and on someone's name, browser's status is starting to work properly. And if I turn of chat there, it will be really off.