Microsoft Manual of Style prefers Log on/logon over Log In/ Login.
Sign in is used by many websites such as Linkedin, Microsoft, and so on. From my experience, log on is mostly used in enterprise documentation.
For example, the information you use to sign into your email is your login (noun), and the page where you sign in is the login page (adjective).
Two words as a verb, one as a noun.
I personally prefer to log on to log in but both are fine.
When we use it as a button in the UI, should it be two words or one word?
Use the phrasal verb form: Log in. The button is offering the user the option to perform an action, that is, to log in.
If this is not the case you can talk to your UI developers.
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