No, it’s retained earnings ending balance = retained earnings beginning balance + net income - dividends

Nothing about dividends vs buybacks has to do with where the cash comes from.

Well, all I can say is get a new dictionary, because companies really can and do borrow money to pay dividends.

Of course, any bailout would have to have terms preventing this, and it should prevent immediately spending it on stock buybacks too.

This is clearly false: there are plenty of value companies with high volatility of profits and low volatility of dividends. Certainly I have seen loss making companies still pay dividends.

No, the definition is a payment from a company to stockholders that's labeled "dividend". I know of companies that were losing money that still paid dividends.