The first, Barton says, is the prayer of quiet trust (Psalm 131). This brings to mind the verse in Psalm 23 that says "he makes me lie down in green pastures." In A Shepherd Looks at Psalm 23, Keller says that sheep won't lie down if they're hungry, fearful, or uncomfortable. The point, then, is to quiet myself, to ask God to bring me to that place of quiet trust, like a contented child.
The second is the prayer of indifference, the attitude of Mary when the angel tells her what's about to happen to her (Luke 1:38 "Behold the handmaid of the Lord; be it unto me according to thy word.") or Jesus in the garden, when he says, Yet not my will but thine be done.
In My Utmost for His Highest, Chambers says, "To be certain of God means that we are uncertain in all our ways, we do not know what a day may bring forth." Which is a frightening concept, but doesn't the Bible already tell us that (Proverbs 27)? And Jesus told us about this too: "If anyone wants to do God's will, he will know whether I'm speaking from God, or just making all this stuff up" (John 7:17).
I need to be in a place where God's will is what I want—the place of surrender. This is really hard, because I want what I want, and I think it's a good idea. If I thought it a bad idea, I wouldn't want it.
But I need to lose the attitude of "Be reasonable; do it my way!" and instead pray sincerely, "Thy kingdom come, thy will be done on earth, and in my life, and in particular regarding this decision." This is something I can't do on my own; I need God's help in getting to that place.
The third prayer mentioned is the prayer for wisdom (James 1:5, etc.); we sometimes go straight to this third kind of prayer, without first getting peaceful or surrendered. When that happens, I may not be trusting God and I may experience unrest, because I don't really want God's will; what I want is for God to give a thumbs-up to my will.
Sorry to say, I have a great deal of experience in this area. I mean of asking for wisdom without first trusting God and surrendering to him.
If I'm considering alternatives, or arguing for why my chosen alternative is better, there's a temptation to have a perfunctory prayer and start writing down advantages (of my way) and disadvantages (of yours). And therein lies the problem: when I want my will, I can come up with all kinds of "reasons" why my chosen alternative is "better" than the way you prefer. I can be very convincing and sincere about it too—I've convinced myself that my way is better, and I'd never lie to you about something unless I'd lied to myself about it first.
Of course I can't tell I'm doing this at the time. But I have seen very smart people come up with really lame reasons why thus and such is a good idea, when it's clearly not.
Quiet trust first, then surrender, then asking for wisdom. May the Lord remind me about this sequence when I get the order wrong.