The definition of "child with special needs" that would apply, for adoption credit purposes, would be what your particular state determines as special needs.
As you suggest, a child with special needs for purposes of claiming the adoption credit doesn't necessarily correspond with the definition used for other purposes.
According to Adoption of U.S. children that a state has determined to have special needs in the IRS' Topic No. 607 Adoption Credit and Adoption Assistance Programs, "[g]enerally, "special needs adoptions" are the adoptions of children whom the state's child welfare agency considers difficult to place for adoption." This also reflects what you're stating, but it also indicates that the definition can vary from state to state.
The Child Welfare Information Gateway, a service of the Children's Bureau, Administration for Children and Families, U.S. Department of Health and Human Services, has published a listing of specific factors or conditions that each state considers to determine that a child can't be placed with an adoptive family without providing financial assistance -- that is, each state's definition of special needs -- on its Adoption and Guardianship Assistance by State page.
There is no indication of when the page was last updated, but it's a U.S. Department of Health & Human Services site, so it should be reasonably reliable. There also appear to be a resources on the site that may be of value to you.
