To sync accounts between devices and workstations, you should investigate iCloud Keychain Sync. In addition to syncing keychains (as expected) it will also sync any "Internet Accounts" across devices bound to the same iCloud account. This includes (in my case) two Macs, one Hac, an iPhone 5 and iPad 3rd Generation (WiFi).
The command to
"Synchronize all accounts" actually is performing an IMAP folder sync operation across all active accounts, not synchronizing account settings.
Additionally, if you use MFA (multi-factor authentication) via an "application password" generated by your service provider, iCloud Keychain Sync will sync the same password credential between your devices. May be best to revoke them all and issue a new one, or pick one from the device you are starting with, and revoke the rest upon completion.
Sidebar: I recommend doing this one device at a time because you may end up with duplicate accounts on some devices while it propagates. You cannot selectively sync account information to only some devices without also losing Keychain Sync, but you can enable/disable "Internet Accounts" selectively on a device after it arrives there.
Bonus -- if you use MFA, you can use invalid passwords to help you identify which accounts are new and which accounts are old. e.g. once you expire firstname.lastname@example.org/iPad for your "Test.com IMAP" account, if you have two "Test.com IMAP" accounts, the one that isn't disabled from a failed login is the one that iCloud Keychain sync gave you.