When I try tapping unplugged, my fingers vibrate the half of the string that is fretted between the fret and the bridge, but also the other half between the nut and the fret one back from the one I am tapping...sorry if this is confusing but I need to know how can I prevent the unwanted tapping sound from the other half?
That's called "harping." You just need to mute the nut side.
Just pull an elastic band over the head and down to the nut and put a facecloth or something in between the rubber band and your strings on like the first fret. Instant string damper!