Permissioned in that case simply means private (you need a permission to participate). It does not mean non-distributed. It also doesn't mean any entity with pe

Permissioned in that case simply means private (you need a permission to participate). It does not mean non-distributed. It also doesn't mean any entity with permission can add anything and its accepted. That would just be a shared/synced database. A DTL still has rules and all participant collectively enforce them. Double spend is still prevented by the same technology. There must be some form of central authority to make it private but DLT limits the power of that authority, it can maybe add or remove participants and such stuff but it can not change the rules. Every participants would have to agree to run the new rules This is certainly useful to connect entries with shared balanced like banks who cooperate. Every bank still wants to validate for themself and not blindly trust whatever another bank writes on the ledger. Moving balanced form someone else or double spend is not possible in such a system, even trough all have the same permissions. A permissioned/private "blockchain" is still a DTL as long as the consensus enforcement is still distributed. If all participants have to trust all others to simply follow the rules then its not a DTL anymore.