I have created a customer account for testing.
If I login to the account I am taken to the account area, if you then go to the homepage it is showing that I need to login although I am still logged in. The My Account tab is replace by the login tab (the my account tab shows if you refresh the page). If you go to the menu page from the home page the My Account tab shows again.
I think I mentioned in a previous post that when you login you should stay on the page you logged in from. If you wish to go to your account area you can use the tab, top right.