Just not to let this go unanswered: The correct solution would be to provide a custom translation to the "unity-greeter" language file and replace each locale's rendition of "Login" with the same locale's rendition of "Start" for the login button text.
I did not do that, however, and frankly I don't know how to do it either. Instead, I actually ended up patching
/usr/share/locale-langpack/da/LC_MESSAGES/unity-greeter.mo. This is a not 100% satisfactory hack, but it did work and bring me onwards, and it did get the users a "Start" rather than "Login" button.
It conceivably might be easier/more obvious to find out how to customize these GUI texts.