Is the aforementioned Hadith regarding the stages of raising children (birth-7 years, 8-14 years, 15-21 years, and thereafter) correct in meaning and attribution to the Prophet Muhammad, may Allah's prayers and peace be upon him, and does it have references?

The circulated narration: "Play with your child for seven [years], discipline him for seven, and teach him for seven, then let go of his reins" is not a prophetic Hadith. Rather, it is from the words of Abd al-Malik ibn Marwan. Sheikh Ibn Baz confirmed that it has no origin, and what is authentically preserved from the Prophet, peace be upon him, is his saying: "Command your children to pray at seven, and beat them for it at ten, and separate them in their beds."

As for the phrase "then let go of his reins," it can be understood to mean granting children independence in their social, academic, and practical decisions, without neglect or failing to offer advice, for righteousness and kindness do not necessitate stripping away freedom and personal independence.