>* the article still loses me because it defines transactions one way (the edges) and then seems to make this big switch that each edge/transaction is really two transactions suddenly (one on each side of the edge)

The author messed up by charactizing edges as one way transactions because in that moment he is only looking at the movement of cash. When he switches to edges being two transactions, he is now recognizing that the exchange of cash leads to the recognition of equities and liablities.

If you give your startup $5000 in cash, you expect to have $5000 in equity on the company books. When you charge to a credit card to buy food, you spend $13 buying food and paid $5 to your creditor so you still owe $-8 in liablities.

>Paying my landlord is "obviously" a transaction from my banking account (negative) into their banking account (positive). How it becomes four transaction is,the magic bit glossed over.

The magic of double entry is that you are only taking into account how your transaction affect your own balances. Thats why you need two entries, If you only debit your own account for rent, you would have a non-balanced set of books which would indicate that something is really wrong. So that why you have to credit Rent expense which is an account that doesnt track an actual balance in your bank account but a running total of all your rent expense for the year.

In the wikipedia example, you have your own set of accounting books and the landlord has his own set of books. That is how there are four entries, but you as an indivdual would only see and deal with your own double-entry.