"In fact there's no list of passwords listed any more. They've gone."
You were not using lastpass correctly then. As stated passwords stored in the cloud, I use lastpass on all my machines work, home, mobile (blackberry, kindle). I create a new account somewhere or change a password at work, I get home and there you go password updated or site listed, etc.
My passwords are all different random 14+ characters - no way I could remember them all. I remember my lastpass password, 23 Characters long.
If you want you could even setup 2 factor auth to get into your passwords. Your passwords are encrypted locally before being sent to the cloud, lastpass has no access to them. So unless you don't trust AES-256 whats the issue?
Use the tool correctly and it works great. And no its not the same as storing your passwords on a piece of paper next to your computer in clear text
As to keepass -- same concept, just encrypted file not in the cloud. Which is one of the best features of lastpass - I have access to my passwords on all my devices in a secure manner.