"More than half" means counting days. Since 2020 was a leap year and had 366 days, more than half means the child must have lived in your home 184 or more nights.
Children are treated as living with their parents even during "temporary absences" which can include boarding school, college and summer camp. Your situation is tricky because an argument could be made that the child was only temporarily away until July when they made the decision permanent and presumably picked up their stuff. But it depends on the facts and circumstances. If the child files a tax return and does not check the box that says "I can be claimed as a dependent by someone else" (some people call this "claiming independence" but that's not technically correct), and you claim the child as a dependent, that will create a conflict at the IRS. They will send letters asking each of you to prove your argument, and I have no idea how they would resolve the issue in this particular case. Good luck.
