make the 'X' close the chat rather than minimise it
'X' is universally recognised as a symbol meaning "close". Most users are going to click 'X' expecting it to close the chat, not just minimise the window. It is not intuitive for the user to have to click the burger icon, then click 'End this chat session' to formally close the chat. What this means is that your chat duration stats are extremely unreliable. If a user chats for five minutes, clicks the 'X', then browses the site for a further five minutes before leaving, their chat duration will be logged as ten minutes, when clearly it should be five. I'd find it significantly more useful if the 'X' closed the chat or, at the very least, if your chat duration stats took minimising into account.
I don't really agree with this improvement. Instead, I'd suggest the following :
Add a – to minimize the chat
Add a X to close the chat