The above is an example of Economist humour. Ha ha.
A recession is when your neighbor loses his job.
A depression is when you lose your job.
Before the 1930s Great Depression, every period of "negative growth" was referred to as a "depression" but after that the term "recession" was coined to distinguish between a mere ill wind and a hurricane.
There's no currently agreed definition of an economic depression, but the definition I like best is where the country's GDP (Gross Domestic product = the sum of everything "made" by the country) declines by more than 10%. Between 1929 and 1933 the US economy declined by 33%.
Debt was a major cause of that depression also - but in that case it was coupled with radical deflation (40-60%) in raw material prices that meant that businesses couldn't afford to service their debt and went bust. This led to a painfully massive 25% unemployment rate.
By contrast the bad years ahead (2008-9) are estimated in the UK to involve a 1% GROWTH in GDP and the chances of a literal economic depression after that are unlikely.
Business isn't going to be easy that's for sure, but let's get a bit of historical perspective here. It's been worse and it always gets better eventually.