I respectfully disagree on your definition of economics: a society that produces only food and shelter has an economy. The fact that this economy seems not to fit in traditional neoclassical views is an aside.

Work is the fact that any human being, in any society, will get up in the morning to get something to eat.

Capital, in any human society, is the fact that this human being will probably not get his food without a tool. That tool is capital. (in all senses ^_^)

Politics is the fact (relatively well established, I think) that human beings are social animals, evolved from other social animals, in a species group (apes) characterized by the existence of coalition of animals that give an edge in the reproduction processes of the individual and are then selected by darwinian mechanisms.

We may prefer to implement liberal social organizations, because our personnal philosophies tend this way or because some scientific aspects have convinced us that societies which value equality or freedom for all are better than others, or because we do not belong to the favorized classes. But this is totally independent from the fact that a democracy or a dictatorship will have to feed their populations to survive as societies, and therefore they will have some economical mechanisms that can be studied.

But all economic systems are primarily value systems. And all value systems are political, in the sense they privilege some individuals and organisations, and handicap others.

I probably agree with that, provided that you allow that some value in intrinsic, not politically dependent. And I add that the (scientific) study of these economic systems is not political by itself. Anyway, the overall economic organization, the fact that some activities or possessions are privileged may be policy dependent, in a low level, diffused, kind of way, as in cultural politics.

What i call a "productive process" is the physical processus that allows food to be found in our environnement (and eaten). You can apply that to other needs that may arise, be it understood that I do not say anything on the nature of these needs, that may be culturally defined (as in: a primitive society living in the rainforest has no deterministic reason to have the same preferred needs as a pastoral society from central asia or an urban society from Europe). The choices between these different kind of societies may be determined by the reciprocal influences of culture on the members of each society. It seems that the occidental model is quite attractive at the moment, as everybody wants to emulate it - with some differences-, maybe for wrong reasons (comfort, life expectancy, sense of luxury...)
It seems also that it is quite predatory, as the regulation (or lack of) we have decided to apply to our economical system since the 18th century is favorizing the accumulation of capital over work or natural resources preservation.

Other societies may have made different choices in the past (like in amerindian societies in the US, where work and preservation of natural resources where privileged). This choice is cultural. and therefore political also, I will agree with you on this: culture is maybe a way to justify some kind of power repartition in society.