Yes, it is possible for a site owner to read the passwords of its users. This is why it's a bad idea to use the same password across different sites/systems.
That being said, it's also possible for a site owner to create the site in such a way that they can't read passwords. WarLight is, in fact, constructed this way. There's no way for me to know anyone's password. They're stored only as a salted hash (it's like an unecryptable encryption.)
Of course, there's still nothing to stop the site owner from changing their code so that the next time you log in, they save the password off somewhere. And generally, you never know how a site is written, so you should never trust that they'll keep your password secure. Even big-name sites like Adobe and high-security sites like banks have been discovered were storing passwords in clear text.
The best solution for most people is simple - use a password manager. You can have one password that you remember that gets you into the manager, and then it stores a randomly-generated password for each site. There's no excuse to not be using one these days. If you need one, try LastPass - it's free and works well. I have no affiliation with them. https://lastpass.com/