You are correct that you do not need to file a Form 1040-X unless you have other changes to your tax return. Each tax year uses a different form 5329, so make sure you use the correct one (2022, in your case).
Wait until you receive the 1099-R forms before you amend your 2023 tax returns. The 1099-R forms you receive will (hopefully) be coded properly and likely won't require you to amend. Since it will be a non-taxable return of your excess contribution plus your growth, only the growth will be taxable.
Another option, but it's too late if you already removed the excess, would have been to reallocate the excess to the 2024 tax year. In the future, it might be worth discussing this option with your broker. In that case, only the growth would have needed to be removed.
[Edited 5/10/2024:8:57 AM PST]
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