Re: Mute doesn't mute music?
This is how they designed it to behave. I agree with you that mute and vibrate should include media sounds, but they assume you want to hear media sounds even when on silent. You will likely have to download an app with a widget that allows you to mute all sounds if you want to do it with one press.
I say likely, because I have a Galaxy Note 3, not an S5, so I can't guarantee there is not a setting on your phone to do this. But if there were it would be something new and unusual. If you don't want to download an app, then you will have to go into the volume settings and set the media to the lowest setting or use the side volume down button to mute and hit the setting gear on the volume slider that comes up to set the media volume.