Nunziante,
No problem for looking into this. I'm still seeing some different behavior than what I think you're describing.
When I login from the Alumni page using LinkedIn, I'm brought to the Home Page. That makes me think the Login URL Redirection in the JLinked Component configuration is set to the Home Page.
When I logout from the Alumni page, I'm left on the Alumni Page. That seems to indicate the Logout Redirection is properly set to "No", leaving me on the same page.
From everything you've said, that's not what you see, but I'm not sure how to recreate it. The only other guess I have right now is to try disabling all caching (in Global Configuration and the System - Cache plugin, if eneabled). Then, clear all cache and test again. Let us know if that changes the behavior at all.
Thanks,
Alex