both login and sign in are the same as authenticate
verb, to validate the user. Otherwise password are not required.
both logout and sign out are the same
log in is to authenticate into a website that has different interface as guest
You cannot use the logged in interface if you are a guest, e.g. Facebook
sign in is to authenticate into a website that has same interface as guest
You can still use the website without signing in, e.g. Google
register is entering your information into the system for your account for serious thing
You need to enter your information like name, address, contact number, etc for domain registration. Another example is company registration.
sign up is to create an account to use
You only need to enter essential information like email, username, password, etc for a Facebook account.