1. to establish as genuine.
2. to establish the authorship or origin of conclusively or unquestionably, chiefly by the techniques of scholarship: to authenticate a painting.
3. to make authoritative or valid.
... which is the way Google strongly suggest it should be in their branding guidelines, for developer and integrators of their Auth and G+ functionality (I learn't this because they(big G) earlier today send me a branding issue notice for the way the G+ button is presented.
Ref: https://developers.google.com/+/branding-guidelines#sign-in-button
They actually use "sign in" -- what you quoted referred only to casing.
However, in reference to their guidelines, note that they allow customization of the button (including text) to fit to the visual feel and to be consistent in an app.