There are a number of "problems" if that is the right description with Joomla's login methodology.
If you set up a menu item or a link to the Login Form [to be found in Site Modules] after logging in a page is displayed with an option for the user to change their password. certian sitations this is unacceptable as the password criteria are displayed to the benefit of any would be hacker.
This is simply overcome by setting the menu item to "Guest".
In order to Logout another menu item needs to point to the Logout Form, and be set to "Registered" or whatever other group has been set up.
Thus is the "trick" to get round hiding the "Edit" profile button. Part 2 of the form is no longer visible. Problem solved.
Another minor problem on the Login Form is the option to "Remember me", and to "Recover Lost Password" and "Recover Lost User Name". This can be arranged with the following css code in the User.css file associated with the template:
/* hide login stuff*/
.login .list-group > a {
display: none; visibility: hidden;
}