This is why I read every single post on this site and watch every video. I've been repairing appliances for 28 years and there is no education like the education of experience. A big thank you to all of the members here. I feel like an even more accomplished tech than I was before finding this site.
Usually I very carefully can peel the old overlay off.. use very little heat from a heat gun. If you "peel" some of the white off I have successfully used a little white out or white touch up paint to cover the exposed areas. If you need a new overlay here it is... but its an expensive sucker... Good Luck!
No, it shouldn't have any continuity from the terminals to the outer casing. If you are getting any reading at all from either one of the terminals to the casing, its shorted. Just like an element in an oven or on a stovetop. Make sure you are isolating the heater from the circuit when testing.
That part in the picture is the turbidity sensor... and should go back where it was on the old pump and motor without removing the wires. The third picture you posted is wiring harnesses that come with the new pump and motor to convert to any configuration.
In reality they are just two pieces of metal that are incased in a plastic housing. One end of the metal is formed so a wire can push on and the other is just bent over in the manufacturing process. You had it right all along.