Place the cubed potatoes in a large pot and cover with cold, salted water. Bring to a boil over high heat, then reduce heat and simmer until potatoes are fork-tender, about 15-20 minutes. Do not overcook them to mush.

Drain the potatoes immediately and spread them in a single layer on a baking sheet to cool slightly and dry off for about 10 minutes. Steam drying helps the dressing adhere better.

While potatoes cool, prepare the dressing: In a large bowl, whisk together the mayonnaise, yellow mustard, apple cider vinegar, sugar, salt, and pepper until smooth and well combined.

Gently fold the slightly cooled potatoes, chopped hard-boiled eggs, celery, red onion, and sweet pickle relish into the dressing. Mix until everything is evenly coated. Be gentle to prevent the potatoes from breaking apart too much.

Taste the salad. If it seems too dry, add a tablespoon or two of the reserved pickle juice until the desired creamy consistency is reached. Adjust salt and pepper as necessary.

Cover the potato salad tightly with plastic wrap and refrigerate for a minimum of 4 hours, or preferably overnight, to allow the flavors to meld. Serve chilled.