Ok here is some of how it is. The top chart is a ref chart but it does not allow for style so since you have beersmith, use it.
Priming sugar is measure by WEIGHT. Do not confuse this with fluid ounces or volumetric measures. If a recipe calls for 4 oz it is not calling for 1/2 cup. Also, the top chart does show that you would use LESS table sugar by a smooch than corn sugar, not more.
With Beersmith, make sure you type in the coreect CO2 volume you want for your style. For most styles I use between 2.6 and 3.0 volumes. (Your example has 2.2 volumes. This is on the lower of most styles I should think.) Also I use 70 degrees as my temp since this will be my conditioning temp for all of my beers.
And yes I realize I am in disagreement with the Palmer chart that says the lower the temp the lower the carb needed. I disagree. Unless that is referring to serving temp. In thatcase there might be a kernel of truth. Until you get experience with the resultant carb level just pick a mid ground carb level like 2.6 volumes and Beersmith will tell the weight of the sugar to use.
Since you don't have a scale, get one. JK a little. Use 3/4 cup of table sugar and get a scale.