Constants are especially interesting because they can be injected into the .config function of a module. No other provider can be injected for the very good reason that .config cannot inject transitive dependencies. Because constants cannot have any dependencies (unlike services and factories) and do not change (unlike values) they are safe to inject during the config phase. FYI, the .run function is invoked (and its dependencies injected of course) after all modules have finished loading and all config functions have been run. If your provider needs another provider as part of its init then you can set that up in the .run function.
Remember that DI in angular is not magic. It is quite literally the creation of a closure whereby angular invokes your provider function with your dependencies passed as arguments.
The difference between factories and services has a lot to do with how they will get used and singularity of them.
I think the mistake of the documentation is that it leads the reader down the path of the convenience functions first rather than instilling a deeper understanding of the core DI mechanism.
As always with Angular docs, they are dense and it always pays off to understand what you are reading as deeply as possible. Use the force, read the source.