Photos from Messages not Syncing in Photos App

My old iMac (Late 2015 running macOS Monterey 12.6.5) is signed in to my iCloud account. My newer Mac Studio (2022 M1 running the latest macOS, Sequoia 15.2) is also signed into my iCloud account. My iPhone 15 Pro (running the latest iOS 18) is also signed into my iCloud account. Within the Photos app in all three devices, iCloud Photo Library syncing is taking place as expected except with the old iMac in a peculiar way — it contains photos and videos that were saved from iMessage, but the others do not. My Mac Studio and iPhone both have the exact same number of file counts in the library (5,572 Photos, 1,007 Videos). I expected the same on the older iMac, but it has 5,914 Photos, 1,034 Videos. Scrolling through both Macs to find the differences, each of the extra photos in my old iMac contains the iMessage icon overlayed on the bottom left of the photo/video. That’s all fine and well, but they aren’t syncing to my other two devices (unless, actually, they aren’t REALLY part of the library and are just being shown in the iMac’s photo library — it just now occurred to me to see if they are present on iCloud.com). I’m trying to migrate away from the old iMac, but wanted to first make sure that I have all my files on the new Mac first. Why am I seeing this difference, and how do I resolve it?