or can outside forces (like the moon or the redistribution of ice/water
masses) also
alter the distribution of mass inside the earth ?

Only very, very slightly.

But I read that even magma streams beneath the earth's crust are affected by
this.


Similarly, redistribution of water (and air) changes Earth's mass
distribution enough to be detectable, but in absolute terms the effect is
extremely small.

Then why is it that affect the mass of the moon has is big enough to force
the earth's rotatational center away from its gravitational center?
Shouldn't that affect the streams of matter distribution inside a liquid body
such as the inner earth?

Solar tides are a bit stronger when the Earth is closer, but otherwise no.

also read that moon vulcanism is caused by these grav tides.

There is no known vulcanism on the Moon today.


But I did read on multiple occassions about eruptions of hot gases having
been monitored on the moon. Those gases where traced back to pockets
of hot rock having been heated up by grav tidal forces.

I actually have it detailed in a German atronomical journal.

Regards
Morenga