Each of these terms has a relative time and technology context in order to understand them better. "Sign In" refers to a person actually signing their name in a book such as at a hotel, or on entering a business. "Log In" refers to the entering of a persons identity in a "log book" for monitoring ingress and egress from a facility. A person may be "logged in" by someone else. For example, if you are with a group of persons. All of the names of persons in the group may be entered by the leader in the "log book" ...they may all be "logged in" by the leader. Sign up is, in my understanding, an abbreviated form of "registration". The "registration" process usually requires the input of additional information. Sign up, implies "signing" your name and making some level of commitment to do something. Today, these two terms are used nearly synonymously.