Economics is what exactly?

Reading the Prologue to Roger Backhouse's book "The Ordinary Business of Life: A History of Economics from the Ancient World to the Twenty-First Century" (a very readable book even for non-economists) I found a number of definitions of economics.

Backhouse offers us

• Economics is the science which studies human behaviour as a relationship between ends and scarce means which have alternative uses. (From Lionel Robbins)
  
• The study of mankind in the ordinary business of life. (From Alfred Marshall. Guess where the title of the book comes from!)
• Economics deals with the production, distribution and consumption of wealth. (I guess that would capture much of classical economics.)
• How production is organised to in order to satisfy human wants. (Surely economics is more than this. What of public choice for example?)
• The logic of choice. (But what about production?)
• The study of markets. (But what about all non-market institutions, like firms or clubs or farms or ....)

I'm not sure any of these really work for me.

Later Backhouse writes

Is is possible, for example, to have societies in which money does not exit [...], in which production is not undertaken by firms, or in which transactions are undertaken without markets.

I'm willing to accept this, apart from the no markets bit. I can't think of an example of a society without markets and I can't help thinking if there is such an example it would be a very poor, very small and very limited kind of society. As the division of labour is limited by the extent of the market, no markets means that the division of labour would be, basically, non-existent. This alone would make for a very poor society.