Perhaps there should be "Mute" and "Mute Everything." Not sure how the UX would work for that - a slider for all three values could be tricky - but it would make it clear exactly what kind of muting you want.
There is "mute everything": turn off your phone completely and it will get rid of all sounds - including alarms which won't go off unless the phone is turned on.
My phone rings the alarm even if the phone is turned off. That makes sense to me, I turn my phone off at night.
The article was talking about the iPhone, hence my suggestion being for the iPhone which does not ring the alarm when it's off. At least it didn't until 3.0 where I tried it last time.
The other option (on the iPhone) is to turn the volume down to zero. Even alarms won't sound then.