Q: In my opinion, when the Silence setting is "only when the device is locked", it contradicts the settings you set above, like Manual. Can someone explain how these settings work together?

You should have seen information in the support document that you linked from. It used to be with the first version of Do Not Disturb that it would only function when the device was locked. That changed with iOS 7, and not it can be active all of the time. With it on Manual, it is on, but if you have the device "awake" that you can still receive calls/texts, etc. Sort of like you have the phone awake, so it isn't a problem for you to receive things. If you set the settings for Always, this means you can be doing things on the phone and not be disturbed by anything else. Does that make sense?