In the OP, double-entry accounting is taken to mean that the two parties to a transaction each keep a record of it (and by extension triple-entry accounting is if three parties keep a record of it). This is a mis-use of the term "double-entry accounting" to mean "two-party recordkeeping".
Double-entry accounting means that for each transaction, there will be two (sometimes more) entries in the ledger (set of accounts) - one on the "credit" side, and one on the "debit" side. For example, when you withdraw $100 from your bank account, from the bank's perspective it is crediting an Asset (Cash) and debiting a Liability (Customer Accounts), so a credit entry of $100 would go into the Cash "account" and a debit entry of $100 would go into the Customer Accounts "account".
Triple-entry accounting implies that a third entry would go... somewhere else in the ledger; but OP isn't describing a classic set-of-accounts ledger.
Double-entry accounting means that for each transaction, there will be two (sometimes more) entries in the ledger (set of accounts) - one on the "credit" side, and one on the "debit" side. For example, when you withdraw $100 from your bank account, from the bank's perspective it is crediting an Asset (Cash) and debiting a Liability (Customer Accounts), so a credit entry of $100 would go into the Cash "account" and a debit entry of $100 would go into the Customer Accounts "account".
Triple-entry accounting implies that a third entry would go... somewhere else in the ledger; but OP isn't describing a classic set-of-accounts ledger.