Cover eggs with cold water by 1 inch in a small saucepan; boil. Cover, remove from heat, and rest 12 minutes. Chill in ice water 5 minutes; peel and chop.
Meanwhile, cover potatoes with cold water by 1 inch in another saucepan; add cooking salt. Gently simmer 8 to 10 minutes until tender but corners intact.
Drain; spread in a shallow bowl and sprinkle with vinegar. Turn gently; cool 15 to 20 minutes until no longer warm.
Mix mayonnaise, mustard, relish, dressing salt, and pepper in a large bowl. Gently fold in cold potatoes, eggs, celery, and onion without crushing.
Cover; refrigerate at least 1 hour. Serve cold, keep chilled until serving, and promptly refrigerate leftovers.