My guess is that the TaxBook just took the examples from Pub 596, because when have tax book publishers ever gone out of their way to come up with original material?  This whole tiebreaker issue requires some deep digging for those who don't regularly come across these conflicts (or agreements).  I would, however, consider it due diligence if the facts matched those in this Pub 596 example:

Example 10—Unmarried parents. 

You, your 5-year-old child, L, and L’s other parent lived together all year. You and L’s other parent aren't married. L is a qualifying child of both you and L’s other parent because L meets the relationship, age, residency, and joint return tests for both you and L’s other parent. Your earned income and AGI are $12,000, and L’s other parent’s earned income and AGI are $14,000. Neither of you had any other income. L’s other parent agrees to let you treat the child as a qualifying child. This means if L’s other parent doesn't claim L as a qualifying child for the EIC or any of the other tax benefits listed earlier, you can claim L as a qualifying child for the EIC and any of the other tax benefits listed earlier for which you qualify.