You are right that there would traditionally be separate books. The first, a "day book" would be a list of transactions that a merchant (or his servant/employee) would write down as business took place. This would say something like "I sold 10 spools of yarn for 8 florins apiece to Mr. Lucio" (remember that this all started a system to keep track of the affairs of Florentine Renaissance merchants). Later, after business had concluded for the day, the merchant (or his bookkeeper) would "post" the transaction to the ledger, which was a book that had a page for each "account." He would go through the daybook and, for each transaction, post it to (at least) two accounts, debiting and crediting equal amounts so that the books "balanced." This left him with a single number for each account which represented the total net activity for that account. This is the account's "balance" and is a very interesting piece of information for a merchant, so much so that it warrants all the associated ceremony.
Today, the "posting" can usually be done by a computer. There is some human creativity/regulation involved in deciding which specific accounts to credit/debit, but this can be noted in the "daybook entry" (more likely a database record somewhere than something literally written in a book) instead of being put off until the posting step.
The important characteristic is what the grandparent noted, that each transaction (in the daybook) tracks completely where each cent (or florin) came from and where it went. If a system of bookkeeping has this characteristic, I think it can be called double-entry, at least in spirit. A hard requirement that a transaction occur between exactly two accounts makes some kinds of events difficult (or impossible) to model, but is still conceptually the same.