What's GDP - really?

"...lies, damned lies, and statistics."

Wikipeadia (so it must be correct!) says,

The term was popularised in the United States by Mark Twain (among others), who attributed it to the 19th-century British Prime Minister Benjamin Disraeli (1804–1881): "There are three kinds of lies: lies, damned lies, and statistics." However, the phrase is not found in any of Disraeli's works and the earliest known appearances were years after his death. Other coiners have therefore been proposed, and the phrase is often attributed to Twain himself.

So, why am I using the above quotation to describe GDP  (Gross Domestic Product).  Well, consider this:

I cut your lawn.  In return, you mow my lawn.  No monetary payment is made.  No effect on GDP.

I cut your lawn - you pay me £1. You mow my lawn - I pay you £1. The GDP goes up. The more transactions per person per year, the greater the GDP of a country.  But neither of us is wealthier.  The transaction left us exactly where we were, but the statisticians employed by government can rejoice!  GDP went up!

What is GDP really measuring?  Increased productivity, or output, or profit?  You tell me; I really don't know!