Place the peeled and cubed potatoes in a large pot and cover with cold water by about 1 inch. Add 1 tablespoon of salt to the water. Bring to a boil over medium-high heat, then reduce to a simmer. Cook until the potatoes are fork-tender but not mushy, approximately 10-15 minutes.
Drain the potatoes thoroughly in a colander, then spread them out on a baking sheet. Allow the potatoes to cool completely to room temperature, about 30 minutes. For best results, refrigerate them for 30 additional minutes before mixing with other ingredients.
While the potatoes are cooling, whisk together the mayonnaise, mustard, pickle relish, apple cider vinegar, and sugar in a large bowl until smooth.
Add the cooled potatoes, chopped eggs, diced onion, and celery to the dressing. Fold gently with a rubber spatula rather than stirring vigorously. This technique preserves the potato chunks and creates the distinct texture.
Season with celery salt, paprika, salt, and pepper to taste.
Transfer the potato salad to a serving bowl, cover with plastic wrap, and refrigerate for at least 2 hours, preferably overnight. This resting period allows the flavors to harmonize and intensify.
Before serving, sprinkle with additional paprika for garnish if desired.