When you log in to ucoz now, you are logged in automatically based on your IP address. What the hell?! What if someone has registered for the website in a public PC cafe or something?! First of all - any person at all using that public computer will be able to log in to your personal ucoz webpage management. I've been able to do so myself. They can take your personal webpage and completely screw up all your hard work or delete it.
cake, login is not specified via IP address - it is set via cookies. Apart from however that is why the option exists to select whether you are logging in on a private or shared computer. This determines whether or not the necessary cookies are set to a persons computer so that they can login multiple times using the same uID or whether it is a single-time login.
Secondly, what if someone changes computers? Let's say that I've registered at my friend's house. Now, i'm at home. I try to log in and edit my things, but can't.
Of course you can. It simply means that once you log in the previous session is overidden and replaced with the new active session.
Thirdly, you can't log in to other accounts of yours. You cannot choose who you are logging in as. Let's say two people are sharing a computer. One is the website admin, and one is a user. The people cannot chose who they are logging in as. The admin might end up always logging in as the other user, cannot do anything to stop this, and will have no control over the website.
Again, yes you can. Firstly, most users on a site will be making use of unique terminals therefore this problem only affects a limited user-group. To get around the problem you have outlined however you need to log out of your uID account before closing the browser. So open http://uid.me and select Log Out uID as your log out option.