A comprehensive overview and comparison between model-based and design-based approaches in impossible to fit into an answer on Cross Validated (not to mention that I'm not qualified enough to even attempt that). Having said that, it is an important and interesting topic, so I'd like to try to partially answer your first question as well as to share some relevant resources for further studying.
Generally, terms model-based and design-based refer to approaches to performing statistical inference from data. Since inference implies making conclusions about a population, based on the analysis of a sample, assumptions about an underlying statistical model is what essentially differentiates model-based approach and design-based approach. Despite the presence of the word "model" in one approach and the absence of it in the other, make no mistake: both approaches are based on a notion of statistical model. The key difference is whether a statistical model considered an unknown construct or a known one. In particular, in a model-based approach can be defined as an approach, where statistical model is unknown (hence, the presence of the word "model", as it's the focus of discovery). Correspondingly, a design-based approach can be defines as one, where statistical model is known and the focus is on a study/experiment design.
Returning to the question's snippet, I would say that it essentially describes the same taxonomy, but in the context of the causality aspect. I'm not sure what paper is that snippet from, but I have been able to find a video of a talk, which has a slide with exactly the same snippet. The talk is a the University of Michigan's 2012 Woytinsky Lecture, which is called "Model-Based or Design-Based? Methodological Approaches in Empirical Micro" and presented by Prof. David Card (UC, Berkley). The part, where Dr. Card provides his explanation of the difference between model-based and design-based approaches in context of causality is located between minutes seven and ten in the above-referenced video of the talk. I have found his explanation clear enough. Let me warn you, that after that slot, if you decide to watch further, the talk contains a questionable statement - which I personally completely and respectfully disagree with - about the equality between data mining and machine learning. But, this is off-topic for the current question, anyway. It might also be useful to watch the rest of the talk - in particular, between minutes ten and fourteen, Dr. Card introduces a "middle ground" approach to causality, which resembles the hybrid approach, mentioned below.
In addition that, there are several resources, which I think are relevant to the topic and might be helpful. For example, the paper "Inference, design-based vs. model-based" by Koch and Gillings (2004) as well as related, but a bit less clear (hence, not fully cited) papers: this, this, this and this.
Finally, I would like to share probably the best resource of all that I have referenced so far - a very comprehensive and nicely-written research paper on the topic by Sonya Sterba (2010). In addition to an excellent overview of model-based and design-based approaches (frameworks), the author introduces an alternative hybrid model/design-based inferential framework.
Koch, G. G., & Gillings, D. B. (2004). Inference, design-based vs. model-based. In Encyclopedia of Statistical Sciences. Wiley. doi:10.1002/0471667196.ess1235.pub2
Sterba, S. K. (2009). Alternative model-based and design-based frameworks for inference from samples to populations: From polarization to integration. Multivariate Behavioral Research, 44(6), 711–740. doi:10.1080/00273170903333574