I've looked at the Wonder Boy puppet now.
The mesh shows up like in my sample in all phantom places (where a layer cycle image would be). But there are no grave distortions.
I wonder if this is, because the head is static in Wonder Boy? In my case the head (with mouth and all) assumes different positions. And the face is where the distortions are horrible.
The mess is in the face really. Since I have different head positions the mouth ends up being in different keyboard triggered places. With this come for example 4 different mouth handles in the rigging view. And they seem to be the trouble makers. I aligned the handles at one of the 4 spots and voilà the weird distortions disappear. The mouth shapes still seem to be where they need to be, so moving the handles really seems to impact only the mesh so far - as desired. And it does make some sense, given what other handles do. I will continue to play around with it, hoping that did not create new problems.
If anyone has an insight, or I'm overlooking something, feel free to comment.
What you're seeing is common. I checked several of my puppets and they look similar.
One thing you might want to try is going independent + staple with the parts you don't want to warp. So for example, if the nose is getting warped weird, try making it independent and stapling it to its parent group. This is usually the "quick fix" method to these kind of issues - if it's independent, it shouldn't get influenced/warped by anything else. And then stapling is a cheap way to pin it to other stuff. If you want to staple multiple parts just select them all and use the menu command instead (Puppet > Staple Puppet To Parent).
I'd say try that and see if it helps.