I like all different kinds of potato salad, and they DO fall into distinctly different camps. All, in my house have in common that they do NOT HAVE:
2. green peppers
3. raw onion
Most anything else is fair game.
Basic potato salad that I grew up eating and continue to make probably 80 percent of the time is either red or gold potatoes, peeled, diced and boiled, with a dressing of mayo (Hellman's, please), mustard, sweet pickle relish, garlic powder, onion powder, seasoned salt and lots of paprika. I've been known to add a dollop of ketchup to this; it's pinkish anyway from the paprika and the seasoned salt. Sometimes I'll add boiled eggs; generally not.
Then there's new potato salad -- tiny new potatoes, halved, boiled, sprinkled down pretty thoroughly with white wine vinegar immediately after they're drained. Dressing is, preferably, a light garlic aioli. Additions include diced cornichons, chopped boiled eggs, parsley, marjoram, chives, whatever other fresh herbs you feel like adding.
I make my own take on German potato salad when I cook German: red potatoes are sliced, not peeled, and boiled to just done. Chop up and saute some bacon til almost crisp, add onions. Remove from heat and stir in caraway seeds and a good portion of coarse mustard. Pour over hot potatoes and toss gently; serve warm or at room temp.
Every once in a while I sail off into the unknown. I did a salad once with fingerling potatoes cut into chunks and boiled, then doused with rice vinegar and light soy sauce. Dressing was mayo, ginger, more mirin, and a little sesame oil. Additions were shelled edamame, cocktail shrimp and julienned carrots. Sprinkled the bowl with toasted sesame seeds.
I always cook potatoes for potato salad cut into bite-sized pieces, whether peeled or not, and cook in salted water.