I am frequently hearing of new research finding how bad sit-ups and crunches are (example). So what are safe ways to exercise the abdominal muscles? For example, are planks proven to be safe?
Abs can be sufficiently trained via isometric exercise. I believe the safest way to exercise the abdominals is to use them for their intended function (stabilization) under a progressively increasing load or difficulty.
The co-contraction of the abdominals with the spinal erectors that is required during heavy squats and deadlifts in order to stabilize the torso (http://startingstrength.com/articles/abs_rippetoe.pdf). As your squat and deadlift get heavier, your abs are forced to get stronger.
Stabilization during pull-ups or chin-ups. These will leave your abs sore.
In a rehabilitation setting, when trying to correct posture-related back-pain, isometric co-contraction is suggested over concentric abdominal work, as well as emphasizing the stabilizing role of the abdominals (http://portalsaudebrasil.com/artigospsb/holist006.pdf):
Exercise involving co-contraction of the abdominal and back muscles is also in line with stabilization.
A simultaneous isometric co-contraction of transversus [abdominal] and multifidus [back], while maintaining the spine in a static neutral position, should help re-educate the stabilizing role of these muscles.
Another publication supports the use of isometric abdominal exercises in a therapeutic setting (http://www.pitt.edu/~neurolab/publications/1997/BeimGM_1997_JSportRehab_Abdominal%20strengthening%20exercise-a%20comparative%20EMG%20study.pdf)
Isometric abdominal exercises [...] have also been found to improve low back pain.
Besides the exercises recommended in the article you cite (planks, bridges, leg-lifts, bird-dogs and "stirring the pot"), full-body exercises that rely on your core muscles are also great ways to improve abdominal fitness, and they often strengthen your back at the same time, thereby helping prevent injury instead of potentially causing it.
Some examples of those are swimming (especially dolphin, butterfly and other strokes that rely on the abdomen -- dolphin-kicking with fins on is one of my favorite ab workouts, possibly just because it's so fun), tai chi (choose a style appropriate to your fitness level), surfing (or skateboarding, or slacklining; anything where you have to balance your body) and rock-climbing (if done properly, focusing on balance and technique rather than just arm-strength). You'll be hard-pressed to find anyone who's a regular at any of those who doesn't have a nice strong stomach.
If you're just looking for one exercise to add to a workout, I'd go with planks. The plank and exercises built on it are incredible stomach-strengtheners -- we use those on my swim-team as daily ab-exercises and they're intense. As with all exercises, form is crucial if you want to avoid injury, so double-check and make sure you're doing it correctly.