Auto-open controls how many chats will automatically be shown in the chat area of the webclient. It doesnot modify the librarian's status to receive incoming chats in any way. Additional incoming chats beyond the auto-open limit are automatically minimized in the contacts list and not displayed to the librarian.
Let's say that a librarian has set her auto-open limit to 2 and currently has no chats. The first incoming chat takes up the whole space. When a second chat comes in, the space splits in two to accommodate both chats. When a third chat comes in, the space remains split between the first two chats and a minimized chat icon is shown in the contacts list for the third chat. To see that chat, the librarian would need to click on that contact in the contact list.
Auto-busydoes modify the librarian's status to receive incoming chats.
Let's say that a librarian has her auto-busy limit set to 2 and currently has no chats. She receives a chat. Then she receives another chat. At the point, she hasn't closed (using the red X in the corner of the chat window) either of the first two chats. Her status is set to busy. Busy allows her to continue with the current chats routed to her but prevents any new incoming chats from being routed to her.
If she is the only operator staffing a queue, that queue also changes to busy since a queue's status reflects the best availability of any of its online operators.
Once she closes at least one of the two open chats (minimize doesn't count, it needs to be the red X), her status is no longer busy and she can receive new, incoming chats.