Capitalism is commodity-producing society. In this political economy, the means of subsistence – things we need to survive – are produced for their exchange value and their use value. Socially necessary labour time is the only source of this value. This roughly coincides with your definition, that a commodity must be manufactured in some way.

But the Wikipedia definition is incomplete. At some point in history of capitalism, humans come to fetishise commodities. At this point, even things that are not produced as commodities, such as land (including their minerals, trees, etc) are treated as if they have value and, from then on, are commofied.

It is the same process that commodifies women, meta data, etc. These things are not produced as commodities, yet they are treated as having a use value and an exchange value. One of the ways that this occurs is through financial derivatives, such as potato futures. This allows someone to buy and exchange something that does not yet exist: hence the commodification of everything under capitalism, even things that aren't 'produced' or aren't yet produced.

This is favourably referred to in the Wikipedia article that you referred to, under 'commodification of labor'. A direct reading of Marx and an analysis of the implications of his work reveals the additional argument that I provided about the commodofication of land.

This 'commodity form' is the root of the problem that you identified to begin with, about rent/wages. I have prepared something about that, which I will post soon.