Doesn't that just mean physical contact? Our skin is made out of proteins, is it not?

Not all proteins can exist as prions. (In fact, it's amazing that some can even exist: think about it: a protein that exist in two stable shapes where the shape that does not normally exist in the body can cause the other shape to make the switch!)

Like a seed crystal causing super cooled water to suddenly freeze in seconds?

Yes, like that

The outer layer of your skin is mostly made of dead cells. It's cold, dry, exposed to the elements, constantly discarded, and has the wrong pH for most biochemical processes that are optimized for the inside of your body. It takes a fairly invasive pathogen to break that barrier.

“Protein” isn’t just one thing.

I’m far from knowledgeable in this area, but the model in my head is: the $foo_protein can be folded in at two important ways, $foo_v1 is what you want, $foo_v2 is a catalyst for refolding $foo_v1 into $foo_v2 but doesn’t refold much (or anything?) else.

Anything expressed by DNA is a protein, but not all proteins are the same. Prion infections are caused by proteins that change the same proteins in the same way, like how crystals replicate.

A human analogy might be a disgruntled but charismatic worker who starts spreading a disruptive attitude until morale and productivity plummet in their department. As long as one of that team remains, the issue can still spread. Coincidentally, both the potted plants and highly compensated leadership are unaffected.