When you disable the Logout button, the LinkedIn button hides.. when you enable it, the Logout button and the LinkedIn button displays? I'm not sure why that would be happening, as it's not an issue we've heard of before.
Please also make sure the "Show Connect Account Buttons" setting is also enabled in the SCLogin module. That's what will show the social login buttons to users that are logged in. It will *only* show buttons for accounts the user hasn't already linked to their Joomla account.
So, once a user connects their LinkedIn account to your site, they will not see the LinkedIn button in the SCLogin module when they are logged in.
One last thing to note is that when a user clicks the LinkedIn button, even when already logged in, their LinkedIn account will be associated to their existing Joomla account. At that point, the user will be able to log in with their LinkedIn or Joomla accounts. That's how the profile import feature works.. it makes a connection between the social network and their Joomla account.
I hope that helps explain,
Alex