I think it is hard to explain, the best thing would be to get a good teacher that can watch you live and help releasing the tensions.
To me the most important point is that simply focusing on the thumb alone may not be the right approach. It is the whole pressing mechanism together that must be tuned up to perform the most relaxed way as a whole. I've learnt a nice metaphor from a Fendelkrais teacher: learning to release tension is like to untie a tangled ball of wool. If you loose one side too much, it may tie harder the other side. Sometimes you discover a node that is holding the whole ball and then you suddenly release everything. Sometimes it is wiser if you start releasing each side a bit and check the whole figure before you attack an specific point.
When working out our mechanism, similar stuff may happen. Maybe releasing a muscle too much would over tense other areas as a compensation mechanism. There may be an internal node we are not even aware of, because of a hard built in (bad) habit. The solution may even be outside the fingers themselves, since a more relaxed and secure overall position would allow better internal awareness. In my experience excessive LH pressure is mainly due to bad posture with unstable guitar and stiff wrist. Working on breathing also helps a lot, breathing works both as diagnosis and solution to over tension.
To me another great misconception is that we must use less pressure over the strings, like mentioned in the previous post. After I worked a while with a renowned teacher that removed several tension problems from my technique, I ended up with more pressure over the strings. The reason is that I wanna be sure they won't buzz. The difference is that I can put more pressure with much less effort or less internal tension. So it is not about the final pressure itself, but the internal mechanism that build up the final pressure.
About the question about using thumb or arm to help pressing, I would say none. Thumb is there as a reference, as well as the arm. The arm is holding finger in poistion, nothing more. If you press the fingers with the arm, the arm "weight" must pass through the finger articulation, so you have to tense the finger muscle anyways to hold the arm weight, otherwise the finger would collapse backwards without transmiting the arm force. So when using the arm to press you are actually overloading the finger muscles with the havier arm weight, which is a path to injury. I think the pressure must come from the pressing finger alone and nothing more.
Finally I think working on relaxation and tension release is mainly working on muscle dissociation control. Our body has a natural defense mechanism which is to tense several muscles just in case. When we try to lift up our heavy sofa, we tense the shoulder, neck, back and the whole body even before we start moving it. This same mechanism acts to some degree when we play. Working the thumb and learning to release it is a nice exercise, you increase your awareness of the thumb and it may help a lot, but it may be just a small node within the wool ball.