The little down arrow means the song is currently downloaded, stored on your device. There used to be a cloud icon, for when music was not local. Now it seems the arrow indicates the opposite. At leas on iOS and iPadOS. PC iTunes may still use the cloud. I don’t know.
Compare Settings > Music on your iPads and iPhones. Pay close attention to the Optimize settings, When turned on it will remove less played music local so that it only downloads on demand. How much music is kept on device depends on how much storage that device has available. A 256 GB iPhone with 150 GB free will keep considerably more music locally than a 128 GB iPad with 32 GB free.
I have Optimize turned off on my iPad Pro, and every song I’ve manually downloaded is there local. I have it turned off on the iPad Mini (way less storage) and almost nothing is stored local.
The Sync Library setting also makes a big difference. To keep the library on all devices as close as possible, make sure it is turned on for all devices.
Whether or not you have an Apple Music account can also make a difference, though it should only make a difference outside the Library tab/section in the Music app.
I can’t say much about iTunes. It has been a long time since I last used it. I don’t really know how it behaves these days, and since it is no longer used in MacOS, I can’t even play with it to see.