I am having a weird problem, I think it's a bug than a feature request. 

Let's Talk - When agents are Online

Leave us a message - When agents are Offline

These two are ideal status update we are planning to use in Freshdesk Live Chat when agents are Online and Offline. 

Example 1: Agents were online and due to an quick training session, they mark their status as Offline in Chat. At this time a customer tries to contact Live Chat (which still shows "Let's Talk!") and enters his information. As soon as he click on "Start Chat" the message option turns as "We are not around, please leave us a message". This gets converted into a ticket.

Example 2: After the quick training session agents get back to their work and turn the chat status as Online. But still Chat displays "We are not around, please leave us a message". Customer or the visitor who are already in the webpage needs to refresh the webpage to change the status. 

Example 1, is livable, but Example 2, I think it's a bug. When Example 1 can auto update the status in Live Chat, shouldn't Example 2 work in the similar fashion?