Yes you are correct. The IRS has published rules for determining the custodial parent and they are found here in IRS Publication 501. The basic rule is the parent with whom the child spent the most nights with is the custodial parent. In case of an equal number of nights with each parent the parent with the highest adjusted gross income is the custodial parent. There are even rules on who gets to count nights spent in other places.
The custodial parent then is able to sign a form 8332 to relinquish the claim of the child tax credit for specific years as directed by the courts. The custodial parent should enter at least 7 months for the time spent with them. That selection will be followed by questions on whether there is an agreement/order to allow the other parent to claim the child tax credit. This also allows the custodial parent (if otherwise qualified) to claim the earned income credit, and the childcare credit and file head of household.
The non-custodial parent should enter 6 months or less which is then followed by questions about the agreement/order and who is claiming the child tax credit for the current tax year.
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