The 1098-T is easy. It goes on your return and you claim the tuition credit. If your income disqualifies you, the 1098-T is omitted. It does not have to be reported on either the parent or student return*.
You are a correct, a one time gig 1099-NEC can be reported as other income or it can be reported as self employment. Other income is unearned income and subject to the $1150 standard deduction and is subject to the kiddie tax. The self employed get an (up to) $12,950 standard deduction. The self employment (social security & medicare) tax amounts to about 14.1%.
*The 1098-T is only an informational document. The numbers on it are not required to be entered onto your tax return. However receipt of a 1098-T frequently means you are either eligible for a tuition credit or possibly your student has taxable scholarship income.
If you claim the tuition credit, you do need to report that you got one.
You claim the tuition credit, or report scholarship income, based on your own financial records, not the 1098-T.
