The main problem with following this kind of exercise prescription is that it is obviously not individualised. The exercises may be inappropriate for some people or they may not address areas of weakness and complement areas of strength. With the exception of push ups the exercises are very much abdominal focussed and as discussed there is more to the core than just the abdominals. They are mostly single plane movements in positions that are not very specific to the sport of triathlon. Given that most triathlon related injuries come from running I would expect a core strength program designed for triathletes to involve at least some exercises in a standing position.
Every consideration that goes into designing an effective exercise program for the rest of your body should go into designing a program for core stabilty or strength: individuality and specificity (exercise selection), load, volume (reps and sets), rest periods, frequency, and progression. In terms of exercise selection then I would suggest you work on exercises that require control or movement in multiple planes e.g. a lateral or side lunge would be a simple example. The exercises should be in position and involve movements that mimic to some extent the demands of the sport.
Core stability and core strength are two linked but different aspects of physical fitness. As a general guide core stability exercises should be performed with a low load, a high number of repetitions, a low number of sets and short recoveries. Core strength however should be trained differently with higher loads, a lower number of repetitions, a higher number of sets and longer recoveries. I would also suggest working on core stability first before progressing to core strength.
Finally, the training of core stability and strength does not occur in isolation. It is part of a larger exercise program designed to develop all aspects of physcial fitness required for a given sport. It should be periodised and will have a different priority during different phases of the longer term plan. At times, the core may be the focus of training. At other times, such as during the competition phase, core exercises may be performed for maintenance only while the focus is on race specific fitness and skills.
On to this weekend's sessions then. Our swim is a technique focussed swim and if you have not yet seen this video from Swim Smooth of Jono Van Hazel you should lake a look. It's a great

See you Sunday. I hope you have those CSS times for us! Tim (LFTC Coach)