That doesn't say what you think it says.
dthomas42 is correct. The CARES act provided for the stimulus payments to individuals by adding section 6428 to the Internal Revenue Code (IRC). That section provides a tax credit for the 2020 tax year. IRC § 6428(a) provides that the amount of the credit shall be:
(a) In general.--In the case of an eligible individual, there shall be allowed as a credit against the tax imposed by subtitle A for the first taxable year beginning in 2020 an amount equal to the sum of--
(1) $1,200 ($2,400 in the case of eligible individuals filing a joint return), plus
(2) an amount equal to the product of $500 multiplied by the number of qualifying children (within the meaning of section 24(c)) of the taxpayer.
I call that credit the CARES credit. The reference to qualifying children in (a)(2) refers to children who are qualifying children for purposes of the child tax credit. Thus, if your kid would meet the requirements of being a qualified child for the child tax credit for the2020 tax return you may, if you meet the requirements as a qualifying individual, take that CARES credit. That CARES credit is then reduced by the amount of the advance stimulus payment (what the IRS called the Economic Impact Payment or EIP) the government sent to the taxpayer over the summer.
This means that noncustodial parents who get a Form 8332 from the custodial parent to claim the child as a dependent in 2020 and whose child would otherwise be a qualifying child for purposes of the child tax credit can claim that CARES credit on his/her return for 2020. Note that the child just has to meet the definition of qualifying child for the child tax credit; the taxpayer does not have to meet all the other requirements for the child tax credit.
This raises an interesting problem with parents who swap claiming the child as a dependent every other year. One of them may have gotten the stimulus payment over the summer based on their 2019 return but will end up ineligible for the CARES credit on their 2020 return. The other didn't get the stimulus payment over the summer but will be eligible for the CARES credit on the return. The one who got the stimulus payment but isn't eligible for the CARES credit lucks out in getting extra money he or she should not have gotten because the law does not allow the IRS to get that back. The IRS addresses that in the last question (Q70) of its EIP FAQ page:
Q70: What if a child's parents who are not married to each other both got the $500 for a child - will one of them have to pay it back?
A70. There is no provision in the law requiring repayment of a Payment. Instead, each parent should review Notice 1444, Your Economic Impact Payment, that the IRS will mail to their last known addresses within 15 days after the Payment is made. The parents should keep the notices for their 2020 tax records.
And the parent who didn't get the stimulus payment but is eligible for the CARES credit on the 2020 return still gets that credit.
Not if the other parent has already received it...
Yes, even if the other parent already received the stimulus payment.