Three Thread networks are mentioned, one of which is not detected (shown in your screenshot) but does appear in an error message? Under Settings → companion app → Debugging → Thread how many networks’ credentials are listed?
Unfortunately the iPhone Matter commissioning workflow doesn’t let you pick which Thread network you want to use when setting up a new device, so having multiple options in this list can cause problems. Even though a Thread network is “preferred” in HA, even the docs say your phone might have a different preferred Thread network it may try to use.
My recommendation would be to join the OTBR (zbt-1) border router to the same Thread network mesh as your Apple TV so that your phone only needs to store one set of credentials. To do this, first go to the companion app under settings → devices → thread → configure, and click the floating “Send credentials to home assistant” button. Then you can make MyHome your preferred network, and under the three-dot menu for the ZBT-1 OTBR, join to preferred network.
If deleting unnecessary Thread credentials in the companion app doesn’t seem to be working (swipe to the left in the debugging Thread credentials list), I’ve heard you can use the Keychain Access app on a Mac that syncs your keychain to iCloud.
Looks like you are using the Android version of the Companion App.
BTW, Android suffers from the problem that once it learns its first Thread credentials, it ignores any others you try to give it (i.e. sync thread credentials (HA to phone) doesn’t really work anymore).
Your first posting shows that HA (your HA green) sees a Thread network ha-thread-a146 and it has the “key in a phone” icon which means HA green will try to provide your mobile device with the Thread credentials from ha-thread-a146 network when syncing creds.
Your mobile device however is showing a different Thread network which is named ha-thread-b78d. This is very strange, unless your mobile device is actually connected to another HA server (i.e. not connected to your HA green but to some other instance).
Yes, looks like the Android companion app? It puts the settings in a different place. (without specifics I assumed iPhone app since you have Apple TV. Forget what I said about combining networks, this is not possible without an iOS device.)
Your app screenshot introduces a fourth Thread network name into the mix, in addition to the three from your first post. I’m guessing these were failed attempts to setup OTBR?
The one you want synced to your phone is the one that appears in HA’s Thread configure page as “preferred”. In order to sync it with your phone it needs the key-in-box icon next to it. In your first post, ha-thread-a146 meets both these requirements.
You will probably need to remove all the old / failed Thread network credentials from your phone, which for Android often means deleting all data for Google Play services. After that, try the “sync to phone” button again and you should have only one network saved (the current / correct one).