The lower box where you see "102 people chatting" is not a link to the Teachers' Lounge per se: it shows recently active private chatrooms on chat.SE. Different mods are going to see different things there. If your own site mod room has been active recently, you'll see that; otherwise, you'll see some selection of private rooms that are visible to mods from across the whole SE network, like the unparented TL room.
I don't think this should be changed. It's often useful to see a quick link to your own site's private mod room in that sidebar, instead of a link to one specific mod room (new-TL) on the other chat server, which is already linked from elsewhere in the sidebar anyway.