If you don't receive a 2022 Form 1099-R reporting this distribution, the IRA custodian apparently is treating it as a 2023 distribution. It's still a distribution of your RMD, satisfying your RMD. An RMD is nothing more than a regular distribution that is not eligible for rollover and is subject to penalty if not taken. There is no other distinction between a distribution that is an RMD and one that is not. All distributions totaling more than $10 for the year are required to be reported on a Form 1099-R.
"The distribution was effective the next day December 30." This implies that the distribution occurred in 2022, is reportable on a 2022 Form 1099-R and satisfies your 2022 RMD despite what the rep told you. However, if the distribution is instead reported on a 2023 Form 1099-R, this would imply that the distribution was not effective on December 30.
