I think the tooltip that tells you that you have new messages is a little bit misleading.
Right now, it tells you the number of total unread messages in chat. That is, it says "10" if you have two conversations with 5 messages unread in each one.

I think it would be better that the info actually meant the number of unread conversations. That is, if you have two conversations with 5 unread messages in each one, the number that appears should be "2", and once you read one conversation it would move down to "1".

This, in my opinion, would make it more intuitive. Moreover, this is common in many other different chat applications.