Not sure this helps, but in the CuteChatUser table, the logged in users to the web site do show up.  Looks like this is a table of all visitor names that show up on the site and are logged by the cutechat javascript code.

However, their sessions don't show up in the CuteSupportCustomer table.  I think this causes them to not show up to the livesupport operator client.

p.s., you might want to rename the "PubicProperties" columns in your tables.  Pubic means something a lot different than Public ;-)