A wish and a prayer

[Chrysophylax Dives]

In his W.P. Ker lecture on 'Gawain and the Green Knight' Tolkien writes:

How does Gawain find the castle? In answer to prayer. He has been journeying since All Hallows. It is now Christmas Eve, and he is lost in a wild country of tangled forest; but his chief concern is that he should not miss Mass on Christmas morning. He was

troubled lest a truant at that time he should prove
from the service of the sweet Lord, who on the selfsame night
of a maid became man our mourning to conquer,
And therefore sighing he said: 'I beseech then, O Lord
and Mary who is the mildest mother most dear,
for some harbour where with honour I might hear the Mass
and thy Matins tomorrow. This meekly I ask,
and thereto promptly I pray with Pater and Ave and Creed.'

It is when he has so prayed, and made an act of contrition, and blessed himself thrice with the sign of the cross, that he suddenly catches sight through the trees of the beautiful white castle, and rides on to a courteous welcome, and the answer to his prayer.

Now, compare this with the story in Lothlórien:

'I’d dearly love to see some Elf-magic, Mr. Frodo!’ ....
[Frodo says:] 'I hope very much that before we leave we shall see the Lady of the Elves again.’
Even as he spoke, they saw, as if she came in answer to their words, the Lady Galadriel approaching.

So first, is this not the same thing? A prayer/wish is granted. If so, my real question is how to distinguish a prayer from a wish?

Nb. This question was asked by me recently on another forum and is posted here now for everyone, but most especially for @Ephtariat. My motivations are two: (1) I would like some illumination on the question; (2) if you don't get stuck into another thread then your replies on the Lust thread are going to carry on coming so rapidily that I cannot keep up. (And I'll keep on adding threads until you slow down on the first one!)

[Ephtariat]

A wish may be addressed to somebody, even a deity, but not necessarily. A prayer instead is necessarily addressed to someone, especially a deity. So we may say that some prayers are wishes addressed to other people, because we also have prayers of thanksgiving and praise.

[Chrysophylax Dives]

Well, I definitely feel a bit stupid now, and see why no one bothered to reply to it last time I asked. Still, I do have an intuition there is a genuine question here. Will have a think and see if I can find it.

[Fuin Elda]

I would also argue that a wish is something that can happen through work and actions in general (Children's Wish Foundation for one). Sometimes it is as simple as making sure you are in the right place at the right time. Wishing in my mind tends to be something happens easier or at all but it is certainly possible in the larger scheme of things (as such I find that hopes of Frodo and Sam to be more in line with a wish than a prayer.) They are in Lothlorien guests of Galadriel it seems likely that it's going to happen.

A prayer is less likely to happen it can happen but it tends to be attributed to the benevolence of a higher power that it happens Gawain is lost in a forest what are the chances he is gonna make Christmas mass? Far less likely that it's going to happen without some sort of luck mind you I find Gawain less than healthily endowed between the ears most of the time. So most things with him are a prayer rather than a wish where as someone that has half a mind of where they are going and a basic knowledge of directions it would be a wish...