Live threads is good for what it is but it's not a chatroom and if i'm not mistaken you would need to actually be in that thread to continuously chat in. I think if your after a chat taiga is probably your best option there's also a free version but the pro version is worth every penny and has many more features.
Shoutbox - a usually small chatbox which displays on the forum page normally only on the front page.
Can have variations as a web support chat to vistors (guests) to support sales.
LiveThread - a dynamically auto refreshing thread typically used for real-time observations on an event.
Independent chatroom suite. This has the advantage of a lot more features like webcam, voice, private and group chat, multiple rooms, fileshare. Can integrate the XF database of members for single login. Can be a second script install or hosted on rental.
Sometimes also offers the web support chat to vistors (guests) to support sales.
Instant Messenger. Desktop or mobile app/ client which instantly displays who is online/ logged in and shares files, text, images. Some do webcam, mic. Can integrate the XF database of members for single login. Can be a second script install or hosted on rental.
There are arguments that these detract and drain from the sacredness of forums. I think this is right if they are iesslapped in without care.
But keeping the Shoutbox entry to a char limit actually enriches threads by draining off clutter posts which should be using the Like system or composing sentences! A full chatroom can be time limited on access to support arranged meetings which actually stimulate forum discussion. A Messenger is great for keeping staff in touch and sorting things out fast.
I use both Chatrooms (gchat) and Messenger (24IM free for up to 10 users eg staff). But I choose to use hosted versions so if one of my comms services goes down I have 2 others to rely on. That means I have to duplicate creating user accounts - (name password, email, avatar) but it;s worth it for small boards.