There isn't always a cut and dried answer to how someone connects to the network of cousins or to John Turner because there are several constraints that make tracking ancestry back to John and Patience difficult, where DNA analysis is concerned.
John and Patience died before 1807. This means that there are many generations between us and them. Although we largely get half of our DNA from each parent in each generation, recombination is random. So, it is possible (not likely but possible) for this worst case scenario to occur :
Given that possibility, and the increased likelihood of less severe related scenarios with every generation that passes, it is not reasonable to make the assumption that someone 4 generations from John Turner would have 3.125% of his DNA. They could have none. For this reason, on the John Turner project, we are less concerned with how much DNA a cousin has in common with another and more concerned with research + admixture in shared segments (more on that below). If a person shares a tiny segment with another cousin, and in a largely European admixture, that one tiny segment has Sub Saharan African admixture, the odds of that being from John Turner are pretty great. Is it science? No. But genetic genealogy is sometimes a matter of odds and we're betting on the high odds here.
centiMorgans are a unit of measurement within a segment of DNA. It's a very finite measurement that tells us things like how related we might be to a person by counting how many cMs are involved in a matching segment. Segments have a start and end point. So you might say, I match this person on Chromosome 3 from position 200-1000, which is a very finite measurement.
However, my experience is that depending upon what service (23andme, FTDNA, Ancestry etc) you use, the start and end points for matching segments differ from site to site. So one site might say you match on Chromosome 3 from 200-1000 and one might say 300-900. I'm not sure why that is the case other than each service uses different processes and methodologies to process the DNA data. While this difference might not matter a whole lot if you're talking great grandparents and 2nd cousin, when there are such tiny bits of DNA left to begin with (5-6 generations), this can make a lot of difference. Sometimes, on this project, I've had a need to make a judgement call where it looks like the process might be in error. That is, strictly speaking, not the highest ideal standard. But, you use what you have.
I've encountered the down side of family secrets more times than I can count. An example of this might be that grandma swears that your great grandfather was John and you aren't related to any of John's descendants. So, sometimes, a person believes they are related to John and Patience because of faulty information, by no fault of their own, and simply are not. The converse is also true, especially in this situation, where the ancestor is an African slave in the South and many of us are white, from white families, who's livelihoods and happiness relied on not acknowledging this heritage. This type of secret or myth becomes codified in public record in the form of birth certificates where the parents' names have been changed to reflect the secret or myth or in death certificates where the person reporting did not know the true story and gave the story they knew as truth.
This makes it challenging to rely too much on only family research alone. In the John Turner project, the family tree is necessary to get us into the right ballpark but then I rely on DNA shared between matches to support that research.
How do you look at a person's DNA and determined what came from what ancestor? Largely, you cannot. However, you can combine DNA with family tree research and matches among cousins and triangulate MRCA, which, combined with enough DNA data and family tree research, I believe can get you to the truth.
On the John Turner project, we use triangulation to identify the DNA from each person that most likely came from John Turner. However, we also have another unique characteristic we can use - John's ethnic background. Because his children and their children largely married and had children with white people, the project participants are mostly some mix of European or other non-African ethnicity so Saharan ancestry sticks out a bit in the admixture of participants. We've had good success so far finding matching DNA segments among cousins that also have sub Saharan ancestry in them. While that's not 100% (see recombination inconsistencies above), if the family tree research lines up, if we can triangulate with several cousins AND they all have that same Sub Saharan African admixture on those matching segments, it's a pretty sure bet we know those snps came from John. We cannot use the same process with Patience because we don't know what her ethnic makeup was.