Wrong way: As described in the article - user types in new password for site. Service updates same password to every site.
Right way: User logs into service with unique password, requests new passwords for all sites. Service logs into site, updates password with new randomized gibberish password that only it knows. Does this for all sites with a unique password for each site.
That is of course assuming you trust an online service with all your passwords. Just use a password manager like KeePass and manage it yourself. If you're afraid of your computer being stolen, you can backup its database online, encrypted with your KeePass password so only you can ever access it. The site backing up the database can never figure out your passwords.