This is a subjective question, but as a team you may need to come to an agreement on what is 'right' for your product. Here are some snippets of information gathered for a team discussion.
Sign In/Sign Out
- Familiarity: Jakob Nielson (the 'father' of UX) did a study back in the early 2000s and decided to use sign in/out rather than log in/out. While that was quite a bit ago, in 2010 Lee Munroe did an informal follow up and showed that sign in/out was still more commonly used.
- The only valid argument against this format is that it can be confusing when used next to 'Sign Up'.
Sign Up vs Register or Create an Account etc.
- Sign Up can hold a more legal/contract feel for some users.
- When using 'Sign In', 'Sign Up' is too close and becomes harder for users to parse, causing more incorrect clicks.
Login vs Log In
- 'Login' is the noun/adjective form referring to the form, page or actual credentials.
- 'Log In' is the verb form referring to the action. i.e. "Use your login credentials to log in via the login page."
Personally, I'm a fan of using 'Sign In' and 'Create an Account'.