Lastpass Immigrant Confused By 1Password Variants - Which One Should I Be Using?

1Password desktop is just that a desktop version allowing local access. It communicates with 1Password in the browser (Chrome extension) which allows you to fill the desired information within your browser. The two are companion apps. But they can also function independently for those that need it.

You also have the use of 1Password.com. Depending on your needs you may or may not needs to access 1Password.com regularly. I have gone months without needing to access 1Password.com. However, I am a a desktop + browser user. At present about the only need I have for 1Password.com is the need for admin related activities.

If you do not have the desktop application installed 1Password in the browser will open 1Password.com to allow you to edit and save your items.

Well, I see my.1password.com which opens my vault in a Chrome browser tab and there's a separate 1Password Chrome extension that is more a popup window, and then there's the desktop application. So again, I'm seeing three different interfaces.

1Password in the browser is the chrome extension. You might see the term used interchangeably at times. Most folks who are interested in things like biometric unlock use the desktop app as it facilities the features use when using 1Password in the browser.