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.