Just to clarify:
because a high school isn't an eligible educational institution for claiming a credit
Determining dependent status and determining who claims any applicable education expenses are two different determinations.
For the purpose of determining dependent status, high school is qualified educational institution.
For the purpose of determining what education credits the tax filer may qualify for, high school is not an eligible educational institution.
This is basically covered in IRS publication 17 at https://www.irs.gov/pub/irs-pdf/p17.pdf on page 26 in Table 3.1 under the test to be a qualifying child dependent. Take note it does not define "a student" or the type of educational institution they are required to be in. Not even on page 27 under the "Age Test" criteria.
