Pot Roasting = Braising
This more recent document from the same association, Cattlemen's Beef Board and National Cattlemen's Beef, supports this by using them interchangeably: 3 Simple Steps For Braising/Pot Roasting Beef
I was trying to figure out what the document from your question may have been implying by the cuts the different cooking techniques were recommended for. And I'd guess, because Pot Roasting is applied to more connective heavy cuts, that braising would be a shorter cooking time. That's my best guess.
Even given a standardized recipe and method, cooking times are always only a guideline. Quality recipes always give you a test for knowing when the product is done.
The reason for this is that there are many uncontrolled (at least from the recipe author's point of view) such as, in the case of ribs:
- Exact dimensions of the ribs
- Natural variation in the composition of the ribs
- Starting temperature
- Variation in your particular oven
That said, the variable most controlling over the time it takes to cook ribs, given a standard recipe and method, is the thickness of the set of ribs. A longer strip of ribs (btw, the full set of ribs on one side is called a slab of ribs—country ribs are not normally sold this way) doesn't make much difference, as the heat will penetrate through the thinnest dimension.
Ribs are a complex food to cook as they benefit from low-and-slow cooking methods, which allow the touch connective tissue of the ribs (collagen) to slowly convert to unctuous gelatin. This conversion is temperature dependent, taking many, many many hours at around 140 F, but happening in an hour or two at 180 F.
Note also that country style ribs are a different cut than back ribs (the baby variety come from a younger, smaller animal). Country style ribs are cut with a lot of meat on the bone.
In practical recipes, the amount of time this takes will depend on the exact cooking method such as braising, roasting, or barbequing.
Recipe request themselves are off-topic here at Seasoned Advise, but you can find many recipes easily by googling "country rib recipe". Indivdual recipes will give you a guideline for a basic cooking time related to the method used in that recipe, and a test or condition for knowing when they are done.
For example, this recipe for braised ribs from the Food Network suggests an estimated cooking time of 1 1/2 hour braising, and tells you that they are done when when the meat is tender (which you would test by poking with a fork, or just trying some).
An alternate method, as suggested in this recipe from Cooks.com is to roast the ribs in the oven. They use three-stage method where the ribs are covered in foil in the middle stage, but the total cooking time is about 3 1/4 hours. They carefully describe how you know when the ribs are done: "the meat should just about fall off the bone.".
Note: I haven't tried these particular recipes; they only serve as examples.
The bottom line is, ribs are generally done when they are quite tender, which you can easily tell by poking them with a fork or trying them. How long this takes will depend on what method you used, the size of your ribs, and a myriad other factors, but will be on the order usually of 2-3 hours.
Best Answer
As so often in cooking and life, the answer is a solid “it depends”.
Braising means preparing food with some liquid, in a humid environment, usually after a roasting/browning step to develop roast flavors through the Maillard reaction. There’s just a few rules of thumb as to how much liquid is appropriate, and cooking methods cover a certain temperature range, both slow cooking and pressure cooking are technically braises.
When braising, you want to strike a balance between keeping the meat moist and dissolving the collagen and concentrating and mingling the flavors of meat and liquid. And all that while not boiling the pot dry .
Both approaches are justified, and I wouldn’t overthink it. Do what works best for you and then you can always adjust as needed.