Is my child a dependent of mine if she is a full time college student, not living with me, but did not pay for more than 1/2 her living expenses?

If she was under 24 as of 12/31/2022 - Yes, she is your qualifying child dependent and she should check the box that says ''someone else can claim me'' on her return.   A qualifying child's income does not matter.   'Away at school' is a temporary absence.  Don't count time away at school as a time not living with you.

If she was 24 or older as of 12/31/2022, no she is not your qualifying relative dependent.  You can't claim her if she made over $4400 and is 24 or older.

Who can I claim as my dependent?    I am assuming she meets the other tests (citizenship, student for 5+ months, not filing with a spouse)

Q. Is my child a dependent of mine if she is a full time college student, not living with me, but did not pay for more than 1/2 her living expenses? She also had part time work income over $12,950.

A. No.  

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test.

She cannot be a Qualifying Child  because she did not live with you. She cannot be a Qualifying Relative because she had more than $4400 of income. 

See full dependent rules at: https://turbotax.intuit.com/tax-tools/tax-tips/Family/Rules-for-Claiming-a-Dependent-on-Your-Tax-Return/INF12139.html

If the student lived with you prior to going away to college, and is only away from home for the primary purpose of attending college,  that is considered a temporary absence from your home.  This is usually true even if she lives off campus. Most students, under age 24, away at college, are still considered residing with their parents and qualify as a dependent, under the QC rules.