One definition I found is:

"The social science that deals with the production, distribution, and consumption of goods and services and with the theory and management of economies or economic systems." (source: http://www.thefreedictionary.com/economics)

I think this is an example of todays understanding of economics. But I would like to search for more different definitions.

One more definition I found:

"A social science that studies how individuals, governments, firms and nations make choices on allocating scarce resources to satisfy their unlimited wants. Economics can generally be broken down into: macroeconomics, which concentrates on the behavior of the aggregate economy; and microeconomics, which focuses on individual consumers." (source: http://www.investopedia.com/terms/e/economics.asp)

Or one more: (Google is my friend) ...

"Economics is the study of given ends and scarce means. Lionel Robbins, biography, from the Concise Encyclopedia of Economics" (Source: http://www.econlib.org/library/Topics/College/whatiseconomics.html)

and there are more:

"What is "political economy"? Chapter I, Principles of Economics, by Alfred Marshall.

Political Economy or Economics is a study of mankind in the ordinary business of life; it examines that part of individual and social action which is most closely connected with the attainment and with the use of the material requisites of wellbeing. [...]" (Source: http://www.econlib.org/library/Topics/College/whatiseconomics.html)

Well, all these definitions are not really definitions. They describe some kind of important facts or characteristics of an economy ... but where is the clear definition?

For a better understanding I copy the definition of "Length" from wikipedia here: "In geometric measurements, length is the longest dimension of an object.[1] In other contexts "length" is the measured dimension of an object. For example it is possible to cut a length of a wire which is shorter than wire thickness.

Length may be distinguished from height, which is vertical extent, and width or breadth, which are the distance from side to side, measuring across the object at right angles to the length. Length is a measure of one dimension, whereas area is a measure of two dimensions (length squared) and volume is a measure of three dimensions (length cubed). In most systems of measurement, the unit of length is a fundamental unit, from which other units are defined."

Such a definition ... of economics - is missing, I suppose.

Your question and your answer are confusing. You are asking about a definition for "one part of the economy" (which I am not sure what that term even means) but then seem to quote a bunch of definitions about what is "economics".

Assuming that is what you are after, then one you missed is that economics "is the philosophy of human life and action and concerns everybody and everything." - L.V. Mises. Essentially the whole science of economics boils down to two words - "men act".

Science is a rational attempt to inform us about the laws of the universe, intent only on discovering truth. It does not care about value or opinions, rather it seeks to understand all observable phenomena to their irreducible cause. Similarly, economic science does not value, but it is seeking to understand and provide people with the information and laws of human action and interaction in society. It does this by studying human behaviour as a relationship between scarce means and the attainment of given ends.

Well, the question is, how I can give a proper definition for "one part of the economy" ... and that means definition of the total economy which can be "cut into pieces" by a defined scale - in its physical presence ... not in a given monetary unit ...

I mean, if you have a rural economy ... all products can be defined as some kind of food (as a very easy example) ... and therefore can be measured by ... maybe its caloric content ...

The same question in principle I am asking for economics.

We live by 100% in a real world. Economy does play in a real world. Therefore there must be a physical expression in different dimensions and characteristics.

The question is: What should we use as the core of economic doing?

And you gave the normally given answer: "Men act." ...

I would say: "Doing work." ... which is the physical description of acting.

But this also stretches out to machine work, controlled by humans.

Doing work is everything.

That is the core of economic doing.

That is the unit to be used as the lowest common numéraire.

It is an absolute value - in the opposit of money, which is always relative.