What are "cooked" potatoes?

Usually that would mean potatoes that have been simmered to fork tender, but you could use potatoes that have been baked too. It's faster to dice the potato before simmering, but for some things it actually tastes better (tiny difference) to simmer first, then dice.

Be careful if you dice first, they will overcook and become waterlogged quickly.

If the recipe doesn't specify, you can use peeled or unpeeled potatoes. I almost always prefer scrubbed, unpeeled potatoes.

Don't forget to salt the water!

EDIT: One more little note. 1/4 inch dice is small for simmering potatoes. If you do it that way, get them out of the water just before they are completely tender, carry-over cooking will get them the rest of the way. You might also want to rinse them in cold water to stop the cooking.

The best way to pre cook potatoes is to do a rough chop and cook in boiling salted water until fork tender, not mushy. A 1/4 inch dice is way too small to do that.

Try dicing the potatoes into 1/4 inch pieces, place in a pile wrapped in wet paper towels into a micro safe bowl and microwave for 3 minutes, add time and more dampness if not done yet.

If you have enough time for preparation, just put the potatoes into the pot without peeling them, bring them to a boil and leave for an hour or so. If they are still hard when you prick them with a fork or knife, bring them to a boil once more. this way you can get them cooked gently without much energy expenditure. Cooking them in the skin also preserves more of the nutrients which sit under the skin.