On Chrome at least, Bitwarden has it's own extension popup window with the vault. This is a completely separate web context. There's no way for the open website

On Chrome at least, Bitwarden has it's own extension popup window with the vault. This is a completely separate web context. There's no way for the open website to detect your keystrokes barring a vulnerability in Chrome itself. If you really want to be safe, you should use the standalone desktop apps and skip the browser extension altogether. Doing that empty tab thing probably doesn't protect you from anything.