The only passwords did not echo in Unix was that TTYs were then popular terminals and these printed a transcript of you session including logon were usually discarded. TTYs are not used with web pages so should use <input type="text" ...> rather than <input type="password" ..>
How many edbrowse users visit websites using TTYs and then discard their sessions in public libraries lately?
It is not a website issue. I also often have lost keystrokes in various applications (not just web browsers) on my MacBook Pro Retina running OS X 10.8.2 (with all updates including the EFI one, but it was also happening before the latter).
It seems to occur more frequently when the cpu is busy (under load) and many windows (Safari) are open.
Transient loss of window focus, pty or low level (USB) data loss bug? If its not EFI, it could still be a low level Darwin TTY bug. Happens under load so not related to transient loss of window focus. OS kernel bug?
Run Lynx. If site does not work flame the webmaster.
I have Lynx (and lots of other TTY and some X11 stuff) in Fink. May have to do everything via screen, lynx and mutt if you don't want to suffer keystroke loss.
Can test with these as they do not load the system. It may only be particular kind of load so entering a number of "while :; do :; done &" may not reproduce.
Know a lot about TTYs, a bit about X11, not Aqua + Quartz.