Tom, I have been designing lifting trunnions for over 25 years, and, like you, have only dealt with the 3rd type (cable around neck). I am familar with the 1st two, and know that they are covered in DuPont standards. The advantages of the those would be safety, I imagine, due to the fact that shackles are more secure. If improperly rigged (say by using a spreader bar length not matching trunnion spread), the 3rd type presents some danger of the cable slipping off the end. Here are some points to keep in mind:
(1) Watch out for torsion. Even for the 3rd type, cable friction induces torsional stress in the trunnion-to-shell weld(s) which is proportional to the coefficient of friction between the cable and the trunnion neck.
(2) Beware lamellar tearing, which is always a danger with high loadings normal to plate surfaces. I have always tried to keep stresses at the neck-to-shell weld(s) below 20% of allowable by increasing neck thickness and/or diameter as necessary.
(3) When pads are added, depending on thickness and/or diameter of the pad, it may be advisable to weld the neck thru the pad directly to the shell instead of welding onto the pad surface. A fundamental assumption in the WRC-107 method of calculating shell stresses is attachment rigidity.
(4) Trunnion end plates serve two purposes: (a) to keep the cable from slipping off the end and (2) to keep the end of the neck round under load.
(5) Trunnions are relatively inexpensive compared to the consequences of failure. I always figured a little conservatism on them to be cheap insurance.