§152 which defines qualifying child is a pivotal code section for HoH filing status because its very definition under §2(b) references that code section for who a qualifying child is.
As you said, there'll be times when more than 1 taxpayer may be able to claim a child as a dependent. Prop. Reg. §1.152-2(g) outlines the rules for how that would be resolved. In particular, subsection (g)(3), which addresses how the claiming of a qualifying child would coordinate with other code sections, states (emphasis added):
"Coordination with other provisions.—Except to the extent that section 152(e) and §1.152-5 apply, IF MORE THAN ONE TAXPAYER MAY CLAIM A CHILD AS A QUALIFYING CHILD, THE CHILD IS TREATED AS THE QUALIFYING CHILD OF ONLY ONE TAXPAYER FOR PURPOSES OF HEAD OF HOUSEHOLD FILING STATUS UNDER SECTION 2(b), the child and dependent care credit under section 21, the child tax credit under section 24, the earned income credit under section 32, the exclusion from income for dependent care assistance under section 129, and the dependency exemption under section 151. Thus, THE TAXPAYER CLAIMING THE INDIVIDUAL AS A QUALIFYING CHILD UNDER ***ANY*** OF THESE SECTIONS IS THE ***ONLY TAXPAYER*** WHO MAY CLAIM ANY CREDIT OR EXEMPTION UNDER THESE OTHER SECTIONS FOR THAT ***SAME INDIVIDUAL*** for a taxable year beginning in the same calendar year as the taxpayer's taxable year. If section 152(e) applies, however, the noncustodial parent may claim the child as a qualifying child for purposes of the dependency exemption and the child tax credit, and another person may claim the child for purposes of one or more of these other provisions. See §1.152-5 for rules under section 152(e)."
This Prpp. Reg. issued in 2017 resonates the IRS' long-standing position as explained in Notice 2006-86, which is summarized in the link
@TAXOH provided.
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Still an AllStar
