LiteMage is designed to cache for both logged in and logged out users. You need to identify what block output those welcome message and add that to customized top links block config. If you are not sure, you can also put the whole header block as private block, which is less efficient than identify the exact blocks. In configuration for top.links, put "header" there. then flush litemage cache.
If you need our help, we also provide paid service and can login to your server and check.
