Thanks, I’m still banging my head 
To be more precise in what I need:
The weights initially come from a skin modifier,
It applies weight “from root to other bone”, so the red is at the middle, then the rainbow goes to the left (rainbow from -0.7 to 0), second image below.
If I apply automatic weight after subsurf, the weight is from -0.5 to 0.5 for each bone.
How to apply the weights from the vertex groups (Bone.00 and Bone.01) for the bones when parenting the mesh to the armature?
I tried with Bone, Bone relative and armature envelope and Armature Deform alone (it throws the same result as above).
The vertex groups have the same name as the bones, is there a way to apply them?
Oops, forgot the pic: