Per-word is usually a rough yardstick that's invoked when trying to figure out the flat rate for pieces of various sizes. It lets publications quote 500x for shortish pieces of about 500 words; 1,200x for mid-sized pieces of about 1,200 words, and so on.

I've never known a publication to count the words exactly, once the story is delivered. A finished piece of 970 words or 1,050 words will get paid the standard rate for 1,000 words, without any adjustments.