The IRS makes the determination. However, when a child is paying "their share" of the expenses and you are not trying to make a profit, this is not considered rent. This is simply cost sharing. As you mentioned in your post, this is simply reimbursement for expenses.
If you were trying to make a profit or charging her a Fair Market Rent based on the rent you could get in the area and NOT on the cost of total expenses for your home, then yes, it would be a rental. How it is listed on the SSA form does not determine how it is treated.
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