I like the way you have differentiated between the wish and the prayer. A wish is nicely described and a "pagan prayer" and a prayer would be a wish offered to deity or higher power. A wish is just a hope or desire to attain something and a prayer is a thought, desire, or question offered to a deity or higher power of some sort in hopes of that prayer being heard and be answered by the deity or higher power.

Wishes get granted.
Prayers get answered.

Sometimes people need to do both wish and pray

Prayer's not just wanting something. It goes way beyond that.

How would you feel if your kids only ever talked to you when they wanted something? What if what they wanted wasn't workable with other things in their lives? I just ran into this this evening when my son wanted to go home with his cousin while we were at a family wedding. Between his cousin's family needing to take my niece back to college and visit our grandma in the hospital, and my daughter having a birthday party tomorrow and both my older kids having various things to be at tomorrow, logistically it just wasn't going to work. But boy, was my son mad at me for saying no! He just couldn't understand, or didn't want to understand, my reasons for telling him no.

I said it on another question, and I'll repeat it. God's not a magic vending machine/Santa in the sky, that works if we poke all the right buttons/say the right words/bow the right number of times. He's looking for a relationship. He wants to talk with us, and he wants us to talk with him.

So prayer can be requests, yes. But it's also supposed to be sharing thoughts and observations; questions that aren't requests; even jokes. And a lot, repeat a lot, is supposed to be listening. Some religions call this meditation. You don't have to be in any particular position or location to pray, and it doesn't have to be out loud. He's not going to turn up his nose and plug his ears if your hands aren't folded.

So wishing can be part of prayer, but it's just one small part.

The way I see it, a wish is just something that you want to be granted or do. A prayer is that you are asking for answer between you and the creator.

In my very biased mind, there is seemingly no difference. I am an atheist, albeit open to discussion. Everyone has wishes and most people at least have prayers. Both your wishes and prayers will inadvertently contradict others' wishes and or prayers as well.

I would like to say that having faith in a prayer is useful, but it surely has no greater effect than a god-like placebo would.

I don't have the article (apologies) but I remember from a psychology class that nations with higher levels of belief in god or less intrinsic control of their destiny are susceptible to more distress.

This is merely an opinion and I do not intend to offend any religious people on this site.