Logon is used for a hardware system that starts up when used, like a computer.

Login is used for a software system where I have to enter my username and password.

Signin is used for identification, either physical such as a photo ID, or digital such as OpenID. What differs here from login is that in the case of an ID, I can use the same ID to access multiple sites, buildings, etc.

Edit 1: I should've added a disclaimer that I have no sources and make no guarantee that these are the official usage of the words. The definitions I'm offering about are based on my personal understanding of the usage, and are purely opinion.