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Q. Would this be considered untaxed income that my daughter needs to include on her tax return?

A. No. 

Q. Does it make a difference who the distribution check is made out to?

A. No

Q. Also, would she need to report the distribution as income when she fills out the 2023-24 FAFSA?

A. Not a tax question. You may have to ask FAFSA people.  That said, it's my understanding, that actual distributions are not relevant.  FAFSA  just wants to know how much is in the  account.

Q. I wonder if anyone can shed light on how this distribution will be taxed when we file 2023 taxes next year?

A.  Turning age 24 and/or not being a dependent essentially changes nothing concerning distributions.  As long as the distribution was used for qualified expenses, it is not taxable, and is not even reported on a tax return; regardless of whether the distribution goes to the owner or the beneficiary or directly to the school.

If a portion of it is taxable, the recipient of the distribution will report it on his/her return (see general 529 discussion below line. 

Turning 24 does  not disqualify your daughter from being your dependent, it just changes the rules under which she qualifies as a dependent.  Since you are the owner of the 529 plan, distributions count as support provided by you (not her).  

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  (23 for full time students), a relationship test and a residence test. .

A person can still be a Qualifying relative dependent, if not a Qualifying Child, if he/she meets the 6 tests for claiming a dependent:

1. Closely Related OR live with the taxpayer ALL year
2. His/her gross taxable income for the year must be less than $4400 (2022).
3. The taxpayer must have provided more than 1/2 his support

In either case:

4. He must be a US citizen or resident of the US, Canada or Mexico
5. He must not file a joint return with his spouse or be claiming a dependent of his own
6. He must not be the qualifying child of another taxpayer

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Qualified Tuition Plans  (QTP 529 Plans) Distributions

General Discussion

For 529 plans, there is an “owner” (usually the parent), and a “beneficiary” (usually the student, whether a dependent or not). The "recipient" of the distribution can be either the owner or the beneficiary depending on who the money was sent to. When the money goes directly from the Qualified Tuition Plan (QTP) to the school, the student is the "recipient". The distribution will be reported on IRS form 1099-Q. 
The 1099-Q gets reported on the recipient's return.** The recipient's name & SS# will be on the 1099-Q.
Even though the 1099-Q is going on the student's return, the 1098-T should go on the parent's return, so you can claim the education credit. You can do this because he is your dependent.

You can and should claim the tuition credit before claiming the 529 plan earnings exclusion. The educational expenses he claims for the 1099-Q should be reduced by the amount of educational expenses you claim for the credit.
But be aware, you can not double dip. You cannot count the same tuition money, for the tuition credit,  that gets him an exclusion from the taxability of the earnings (interest) on the 529 plan. Since the credit is more generous; use as much of the tuition as is needed for the credit and the rest for the interest exclusion. Another special rule allows you to claim the tuition credit even though it was "his" money that paid the tuition.
In addition, there is another rule that says the 10% penalty is waived if he was unable to cover the 529 plan withdrawal with educational expenses either because he got scholarships or the expenses were used (by him or the parents) to claim the credits. He'll have to pay tax on the earnings, at his lower tax rate (subject to the “kiddie tax”), but not the penalty.

Total qualified expenses (including room & board) less amounts paid by scholarship less amounts used to claim the Tuition credit equals the amount you can use to claim the earnings exclusion on the 1099-Q. 
Example:
  $10,000 in educational expenses(including room & board)

   -$3000 paid by tax free scholarship***

   -$4000 used to claim the American Opportunity credit

 =$3000 Can be used against the 1099-Q (usually on the student’s return)

Box 1 of the 1099-Q is $5000

Box 2 is $2800

3000/5000=60% of the earnings are tax free; 40% are taxable

40% x 2800= $1120

You have $1120 of taxable income  

**Alternatively; you can just not report the 1099-Q, at all, if your student-beneficiary has sufficient educational expenses, including room & board (even if he lives at home) to cover the distribution. You would still have to do the math to see if there were enough expenses left over for you to claim the tuition credit. Again, you cannot double dip!  When the box 1 amount on form 1099-Q is fully covered by expenses, TurboTax will enter nothing about the 1099-Q on the actual tax forms. But, it will prepare a 1099-Q worksheet for your records, in case of an IRS inquiry.

On form 1099-Q, instructions to the recipient reads: "Nontaxable distributions from CESAs and QTPs are not required to be reported on your income tax return. You must determine the taxability of any distribution." 

***Another alternative is have the student report some of his scholarship as taxable income, to free up some expenses for the 1099-Q and/or tuition credit. Most people come out better having the scholarship taxable before the 529 earnings.