Data Generating Process
A data generating process is the process that generates the observed data. We do not KNOW it, but we can make hypotheses about it.
A statistical model is a hypothesis about that process and we test the model against the observed data to determine how well it fits. If we are comfortable with the degree of fit we consider our hypothesis to have become a theory.
here describes the order of observation and DGP
The types of the data sets are not determined by the visualization process, but by the data generation process
In this case ϵ becomes the residual unaccounted for by the model. It is misleading to label it unsystematic or noise because in most cases what it contains is lower order processes contributing to your desired variation (signal) and contributing to undesired variation (noise). If you dismiss it as noise you will never refine your hypothesis and update your model. If you recognise that it may contain some real processes you hadn't anticipated you can explore it and deepen your understanding.
If the universe is truly deterministic then there is no such thing as random and even the tiniest blip in the DGP is non-random. Rather, it may be caused by the faint ghosts of quantum entanglement just after a particle condensed from the big bang, propagated over billions of years and diluted by interactions with other particles and fields. If quantum mechanics has truly random elements then you can push the DGP back to stochastic processes on a quantum level.
Here the authors discuss randomness in exactly this kind of ambiguous way, as a result of deterministic processes
Randomness and data imperfection are two direct consequences of the dynamic nature of stream data. There could be several unforeseeable factors that affect the processing chain. For example, the data generation process may induce randomness because the data sources are normally independently installed in different environments, which makes it nearly impossible to guarantee the sequence of data arrival across different streams
So then for the specific items requested:
this is the desired variation, often referred to as signal. It is something that can be described succinctly and systematically.
this is any variation that lies outside the specified hypothesis. In a deterministic world true randomness is impossible, but is used as shorthand for stuff that is too complicated to untangle. In a world containing randomness the component will not just be the original noise, but all the events that it has propagated into.
Can we always decompose a random variable into something "systematic" and "random"?
As Pohoua says, this is confusing the terminology -a random variable can be combined with a systematic process in a stochastic data generating process. A truly random variable would have zero systematic contributions, something we can't generate.
Is a "data generating process" the same thing as a "statistical model," and is that the same as a "structural equation" or a "theory equation"?
See above for first part (No). A structural equation (or theory equation) is usually the terminology used when a mathematical model is generated based on theory rather than data and is then fitted to the observed data to test. Here comparing physical models to the DGP is mentioned.
If we know something about the physics of the data generation process, we can use that information to construct a model
Sometimes structured equation modelling is used in the context of regression as it creates a structured equation through statistical modelling, but many don't like this usage.