Peel the potatoes and cut them into 1-inch cubes, placing them in a bowl of cold water to prevent browning. Finely dice the onion into 1/4-inch pieces and dice the celery for extra crunch, setting both aside separately. Pour 1.5 cups of water into your Instant Pot and place the trivet (steamer basket) inside. Arrange the cubed potatoes on the trivet, then carefully place the 4 eggs on top of the potatoes—they'll cook together and the eggs won't crack sitting on the potatoes.
Close the Instant Pot lid and set to high pressure for 4 minutes. Once the timer goes off, perform a quick release by carefully turning the valve to release steam immediately. Open the lid and using a slotted spoon, transfer the hot potatoes to a large bowl. Transfer the eggs directly to a bowl of ice water to stop them from cooking further and make them easier to peel. Let both the potatoes and eggs cool for about 15 minutes—I like to do this step so the cool potatoes won't break apart when mixing and the eggs peel cleanly without that gray-green ring.
While the potatoes and eggs are cooling, whisk together 1.25 cups of mayonnaise, 2 tablespoons of apple cider vinegar, 1 tablespoon of yellow mustard, 2.5 teaspoons of salt, 1 teaspoon of black pepper, and 0.5 teaspoon of sugar in a separate bowl. Stir until smooth and well combined. I prefer using Hellmann's mayo for this—it has the right consistency and flavor that makes the salad creamy without being heavy.
Once the eggs have cooled, gently peel them under cool running water and chop them into bite-sized pieces. Add the cooled potatoes from Step 2 to your large bowl, then add the chopped eggs, 1 cup of pickles (Mt. Olive bread and butter chips work great for their sweetness and crunch), and the diced onion from Step 1. Mix gently to combine without breaking up the potatoes too much.
Pour the dressing mixture from Step 3 over the potato and egg mixture, then add the diced celery from Step 1. Gently fold everything together until well combined and evenly coated. Sprinkle the 0.5 teaspoon of smoked paprika over the top and give it one final gentle stir. Taste and adjust seasonings if needed. Chill until ready to serve, or serve at room temperature if you prefer.