If you clicked Install from the link...of course you have to log in...it's the Steam webstore, not Steam itself.
The link in the website launches in Steam, the password prompt was from the Steam client.
if you're having problems you need to restart your Steam client. Stupid bug but at least it now works.
Steam restarted itself when I canceled the password prompt. It did it three times. It finally downloaded after the 4th try, I guess 4th time's a charm now?
I searched the Steam forums too, it looks like a lot of people have this problem with demos. This is the first time it has ever happened to me, and I've been using Steam since HL2.
"The only way anyone can live in peace is if they're prepared to forgive." - The Doctor