Do all covered pickups have the logo or not?

[Mincer]

No hazing on the plastic afterwards or anything?

In my case, you could never tell a logo was on there. As far as covered WLH, there isn't a logo on the cover, and I've never seen a logo under the cover, either.

[The Dali]

Correct - as Mincer does, I use a little bit of Brasso and some elbow grease and looks perfect. If there is any hazing I will do the other bobbin to match, but in most cases it looks perfect.