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There is something that I can't understand about the "Weight Paint".

I was testing some stuff with a few Models, I need to decimate and fix this one for VRchat, but how does the original model rotating all parts so smoothly?

For example, when I rotate the Head bone of the Original upwards, the head looks nice and it seems to stretch all vertices nearby even if the weight of those are "0". That gives is a smooth look.

When trying that on my version of model with an almost self-made Rig, it only affects the vertices that are painted. This deforms my model in an unexpected way.

I made 2 screenshots for an example:

Original

My version

Why does the head of the original affect all those vertices on the neck with 0 weight and stretch them slightly, while on my version its just pulling on the painted ones?

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A smooth deformation of a bunch of vertices is always due to a balancing between 2 or more different weight assignements: if a vertex is weighted to a vertex group only, it will move with its relative bone rigidly and exactly, even if the weight is 0.00000001. Instead, if it has some weight relatively to two (or more) vertex groups, the movement is calculated making a comparision between the two values, with an automatic normalization; normalisation means that if one value is 0.001 and the other is 0.009, they are treated as 0.1 and 0.9 and their sum will be treated as 1.0.

So it's importany to analyze all weights, in your case not only head bone but neck bone also. If you select a weighted vertex in edit mode, you will find in the "N" properties panel a "vertex weights" tab in which you can inspect all weights and have a better understanding of the behaviour.

To solve your problem you will have to give some weight to the neck vertex group, so that vertices near the neck will be pushed to two different actions: the head bone will try to move them while the neck bone will try to make them stay in place.

Image is 2.79 but 2.81 is the same.

enter image description here

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