Here is the long answer to your choke question. In order to break a trap target, you have to send a pattern of shot downrange to intercept the moving target. Ideally, this pattern should be as wide (large in diameter) as possible, while still maintaining enough pattern density (pellets per unit of area) to prevent the target from slipping through. Yes, it is possible to miss a target, even if your aim is dead on. If your pattern has "holes" in it, you can shoot correctly and still fail to break the target.
The pattern is determined by a number of factors, but is affected most by 1) the degree and type of constriction of the muzzle (choke) and 2) the LOAD you are using. Most people do not think of the load when they are talking about chokes. In fact what you are talking about is patterns, and you must talk about the choke and the load when talking patterns.
Two different loads will not pattern the same way when shot through a given amount of choke. This should come as no surprise, after all rifles do not shoot different loads to the same point of impact, so why should shotgun shells all pattern the same?
Degrees of choke such as improved cylinder, modified or full are not international standards, they are the makers interpretation of what those patterns should be. Depending on the manufacturer, a full choke will provide between 0.35 and 0.45 inches of constriction. You cannot expect that a .35" choke will provide the same pattern as one measuring .45, but both are considered "full" chokes.
12 gauge shotshells for targets may contain 1 1/8th oz., 1 oz. or 7/8th oz. of number 7 1/2, 8 or 9 shot, and may be as slow as 1050 fps or as fast as 1400 fps. Different shells will not pattern in the same manner if shot through the same barrel.
The ONLY way to know what kind of patterns you are getting is to shoot your choke and load combination at paper, at the distance at which you expect to break the target. In the case of 16 yd. ATA singles, that could be 26 yards, for the second shot at doubles that could be 40 yards.
A gun which puts 75 percent of a given load in a 30 inch circle at 40 yards would be an ideal full choke gun, unless it concentrated most of the pellets on one side of the circle, or left a doughnut hole in the middle. Unfortunately, the ideal gun and load combination has yet to be invented, so ALL of us shoot patterns which are less than perfect. Some load / choke combos throw patterns which are much more even and consistent than others. If you spend any time shooting patterns, you will find that the differences are not subtle. You can shoot light modified and extra full patterns out of the same choke tube, just by changing loads. Some loads (such as spreader loads) are designed to give wide patterns out of tight chokes. Loads with hard shot and full length shot cups will often (not always) give tight patterns, regardless of the choke used.
To determine the "best" setup to use at trap, there is no substitute for shooting patterns at paper. Experiment with what loads are available, and try them with different choke tubes. Select the combination which produces the most evenly distributed pattern at your longest shooting distance, and you will be well ahead of the game. The results may not be what you expect.
Sharptail
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