In fact, Jews do not believe that God actually listens to prayer. He might, but then again God might not. That is not what prayer is about in Judaism.
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Instead of believing that one's prayers are being heard by God or one is asking God for something, like Jehovah's Witnesses do, Jews are doing something entirely different. Jews believe they are mostly hearing God talk to them.
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There is little in the way of requests in Jewish prayer. Even when one is sick or one dies, prayer is blessing God, asking that one learns to accept their place in the constant changes of reality, blessing these as they come, etc.
That's fascinating, but if so, that raises questions in my mind.
Does the evidence indicate that was what Jewish prayer always meant, or was that what it came to mean through human scholarly and rabbinic traditions over the centuries? The indications from many specific prayers recorded in the Hebrew scriptures suggest those giving them often requested something personally and truly believed God would hear them, for example:
Abraham - "Then Abraham prayed to God, and God healed Abimelek, his wife and his female slaves so they could have children again" (Genesis 20:17)
David - "Lord, the God of our fathers Abraham, Isaac and Israel, keep these desires and thoughts in the hearts of your people forever, and keep their hearts loyal to you. And give my son Solomon the wholehearted devotion to keep your commands, statutes and decrees and to do everything to build the palatial structure for which I have provided." (1 Chronicles 29:18,19)
Jacob - "Save me, I pray, from the hand of my brother Esau, for I am afraid he will come and attack me, and also the mothers with their children." (Genesis 32:11)
Hannah - "In her deep anguish Hannah prayed to the Lord, weeping bitterly. And she made a vow, saying, 'Lord Almighty, if you will only look on your servant’s misery and remember me, and not forget your servant but give her a son, then I will give him to the Lord for all the days of his life, and no razor will ever be used on his head.'" (1 Samuel 1:10,11)
Some of those verses were mixed in with what your quote calls more fixed prayers which I can understand, but are clearly personal requests or pleadings - for example, David's words in 1 Chronicles.
On the other hand, if it was always understood to be that way, doesn't that mean that the kind of prayer Jesus taught his followers was radically different? Jesus clearly taught that prayer was intended to include direct personal requests, petitions, hopes, fears and desires, as well as the things you mention (wisdom to accept ones' place, thanksgiving to God, recognise his will, etc), and so was not mainly to be "fixed" in the same sort of way.