This is a checksum-like trick for manual record keeping, and isn't specifically relevant to the grandparent. It isn't required in digital world. The grandparent talks about having to write 2 separate transactions in 2 accounts in such a way that either both are successfully written, or none are written. This means that there must be transactionality between 2 services, regardless of balance being cached or not.
The point is that there is no such thing as "2 separate transactions in 2 accounts" in banking. There is one global ledger that contains transaction between all accounts - if you move money from account A to account B, you append one entry to the ledger that says that some sum of money moved from A to B. Then if you need to calculate an account balance, you look at all transactions in the ledger, filter by the account ID, and calculate the total. (You can then cache this result, for performance benefits, but this changes nothing about the way it works.)
This is really more like a CRDT than like transaction control.
It's also where Bitcoin took its inspiration from, the blockchain is just one big ledger with some hashing on top.
This is a good explanation for Bitcoin-type ledger, but as far as I understand, banks don't share a global ledger. In US they run settlements through FED, which is the closest thing to global ledger we got, but internally they keep their own separate ledgers. Fed sees settlements between banks, but allocation within a bank is done via internal transactions. If you wanted to give banks a transfer rail that allows them to communicate directly incurring liability, and later settle via FED, you would need to write a transaction in each bank's ledger.
