. . . Strength against cuts has more to do with sidewall design (how high the tread goes up the side), or type of sidewall material.
But its not like they don't test this:
http://www.nhtsa.gov/DOT/NHTSA/NVS/Vehicle Research & Test Center (VRTC)/ca/Tires/811797.pdf
The following quotes from the Abstract and the Executive Summary in the linked NHTSA report are quite illuminating:
From the Abstract:
. . . Six passenger tire models were also tested using an experimental sidewall bruise/strength test and generated statistically different levels of bruise width, penetration, and rupture force between 1-, 2-, and 3-ply sidewall tires. [Emphasis added.]
From the Executive Summary:
The final goal of the agency research was to evaluate tire sidewall strength/bruise resistance. A sidewall test was proposed that used existing FMVSS tread strength test fixtures in an attempt to duplicate the sidewall bulges or broken cords seen in tires damaged during service. This method was used on five passenger and two light truck tires to examine the concept. The sidewall strength results show a difference between tire sidewall constructions. These results suggested plunger penetration and breaking force were significantly influenced by the number of plies in the tire sidewall. [Emphasis added.]
The NHTSA report supports what Stumpalump, I and several others have been saying all along, specifically that sidewall strength is influenced more by the number and type of plies than by load carrying capacity rating, i.e., that generally speaking an offroad tire with 3-ply sidewalls will have more puncture resistant sidewalls than a similar tire with a greater load carrying capacity rating but only 2-ply sidewalls.
The OP's original question was,
"Are there tires out there that have notably tougher sidewalls than my [Dakota Definity load range E A/T] tires that are more suited to my uses or do I just need to chill out and be more careful?"
The answer is yes, there are tires with tougher sidewalls. The OP's tires are a Pep Boys store brand manufactured by Cooper and designed to meet a particular price point. Although given an E load rating, they only have 2-ply sidewalls. Compare that with the Cooper Discoverer ST/Maxx, also an E load rated tire, but with "3-ply “Armor-Tek3®” sidewall construction consisting of two 1,500 denier plies + one 1,000 denier ply.
The ST/Maxx and other Cooper tires with the "Armor-Tek3®” carcass construction are known for their tough sidewalls. The reason is the 3-ply sidewall construction of the Cooper tires, the thickness of the plies and the materials used.
Load rating is simply a measure of how much weight a tire is rated to carry at a particular tire pressure. As demonstrated by the sidewall failure of OP's load range E tire, load rating alone is not a reliable measure of sidewall strength.