The Microsoft Manual of Style for Technical Publications recommends using "log on [to]" -- but "log in [to]" is equally correct, grammatically speaking. "Log in" and "log out", in your examples, are intransitive inseparable phrasal/prepositional verbs so although it's common to see it on the web, you shouldn't use "log into" because "in" is an essential component of the verb. It's similar to "write in": "Listeners can write in to the station" not "Listeners can write into the station" -- there's no physical movement involved, so you can't write "into" something. "Login" and "logout" are nouns or adjectives, depending on context ("My italki login doesn't work anymore", "After clicking the link, the logout screen appears") and shouldn't be used in place of the verbs "log in" and "log out". Your first option can work if you replace <name of system> with <name of user/account>, so the verbs become transitive and separable: "Log in the user if they're in the httpd group" or "Log the admin out if the session has expired".

The second option is closest: For example: "I need to log into Facebook before I can send a message" "I need to logout of Facebook before I leave the computer" You can also say: "I need to sign into Facebook before I can send a message" "I need to sign out of Facebook before I leave the computer" Both are equally correct, and mean the same thing.