Depending on what you want to do you can override the template for access denied (\design\standard\templates\error\kernel\1.tpl ) Please note that unfortunately there are no override conditions for this, so you can just create on override for the entire siteaccess.
I am not sure what you mean when you say that users with an account have somewhere a message like "Welcome, Anonymous User. _Logout_". If they are logged in they are not anonymous anymore and should be greeted with their name. If they are not logged in they are the same as an anonymous user. EZ cannot know whether a visitor has an account until this visitor logs in. Thus they cannot have a 'logout' link as long as they are not logged in.
Greetings from Luxembourg
Claudia
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