I think that if we look at things all prayers directed towards others are for the purpose that each person will be properly guided. the reason we pray is much more important than what our words are. Intention is probably one of the greatest keys for our prayers to be answered.
Would I pray for a non-Muslim to have good health so that he can attack Islam? NO.
Would I pray for a non-Muslim to have good health so that he may be guided to truth? YES
That's an interesting concept, and one I will have to think about ...
Do we pray for God's intervention in non-believers' lives, because
we have specific intentions for them?
I am not sure ...
Is it not for God to work
his will in people's lives, rather than our will?
I believe Jesus' instruction to us (according to the gospels) was to love, pray and care for our neighbours (and that includes our 'enemies'), regardless of
their attitudes towards
us ...
"You have heard that it was said, 'Love your neighbour and hate your enemy.' But I tell you: Love your enemies and pray for those who persecute you, that you may be sons of your Father in heaven. He causes his sun to rise on the evil and the good, and sends rain on the righteous and the unrighteous.. If you love those who love you, what reward will you get? Are not even the tax collectors doing that? And if you greet only your brothers, what are you doing more than others? Do not even pagans do that? Be perfect, therefore, as your heavenly Father is perfect."
(Matthew 5: 43-48)
Both are essentialy the same prayer, yet totaly different intent.
I am trying to think your statement through with an example:
Imagine you have two non-Muslim neighbours. Neighbour A is positive towards Islam, and you hope may convert at some point in his life. Neighbour B is negative towards Islam, involved in anti-Islamic campaigns, and - in your mind - unlikely to ever consider Islam.
Imagine both fall ill with a serious illness ... would you only pray for A, but nor for B??
I cannot imagine that!
Equally, imagine both are drowning in the river, and it is within your ability to save both. Would you only save A, but not B?
I don't think you would.
On that basis, should we not pray for and help people in all circumstance, no matter who they are and what they do and how they act towards you? Who knows what plans God has for any of those people!
I like your peace prayer.

Perhaps I can share one of my favourite prayers here. It is said to have been written by Francis of Assisi. I don't think it will offend anybody, and I hope it can remain in this thread:
Lord, make me an instrument of your peace,
Where there is hatred, let me sow love;
where there is injury, pardon;
where there is doubt, faith;
where there is despair, hope;
where there is darkness, light;
where there is sadness, joy;
O Divine Master, grant that I may not so much seek to be consoled as to console;
to be understood as to understand;
to be loved as to love.
For it is in giving that we receive;
it is in pardoning that we are pardoned;
and it is in dying that we are born to eternal life.
Peace