Thats good though because now you know the baseline you have to work with. So next time you should do 10 reps, and then rest 10-15 seconds. Then do 10 more reps. Rest again but for 15 - 20 seconds this time. Rinse and repeat. Using that method, you'll probably double your first attempt on the second try.
Personally, I'd be more focused on proper form than quantity and speed. Not that they are necessarily mutually exclusive, but it often works out that way in the real world. For example, the linked article in the OP advises not to sag the hips while doing the exercise. However, the photo shows at least 2 guys with sagging hips during the movement: Number 75 is sagging as well as the second guy from the right in the red shirt. Sagging makes the movement much easier. From personal experience, I know that you have to be very conscious of not sagging at the hips towards the end of the set if you are going all out. It takes effort to hold the proper plank throughout the exercise and it also makes the movement harder, as it should be. The guy at the far right of the photo has it right. http://www.builtlean.com/2011/02/23/how-to-proper-push-up-form/
This is something I know a little bit about. As I mentioned in the recently closed thread, my principal cardio exercise is bodyweight squats, which I had been doing with speed intervals. So I would do 25 at a brisk pace, followed by 25 at max speed and repeat until I did the desired number. However, the max speed got to be a bit taxing after a while, and so, I just do them very quickly, but not quite at full max speed. I do 300 following my resistance workout and then immediately follow with 2 ab exercises, which I regard as an extension of the cardio. And on the weekend, I do a set of 400 without the abs. My best time averaged just under a second per rep, but most times I exceed the one second per rep by a little bit (i.e. 1.05-1.10 seconds per rep). I should point out that I only go to just below a 90-degree angle at the knees rather than all the way down or as far as I can go. I initially tried going lower, but it slows the rep down and takes away from the cardio component. Also, I'm less likely to inadvertently pull a muscle bouncing at the bottom at speed. I like bodyweight squats more than, say, the treadmill or the StairMaster, because of the better range of motion. Plus, I don't have to wait to use any equipment at the gym; I just need a mat to quickly slip into abs after the last bodyweight squat rep.
Did the challenge on Monday. Tried a different rep/rest cycle to change it up. Only got 84 this time. Had solid muscle soreness Tuesday and this morning so I know I still hit the pecs well. Finding the ideal rep/rest cycle for your body is easily worth 10-20% more total reps.
I did do something similar when I was in better shape a year ago. Better heart shape. I took an 18lb. ball and held it chest high and did squats. 15 reps, rest 20 seconds. Do 3 sets. Then I would immediate take that same ball and do Wall Ball sets of 15 with 20 second rests. Then I would immediate do 3 sets of 15 Kettle Bell swings with 30 lbs. 20 second rests. That was one cycle. I'd rest 2 minutes and repeat the process for 3 full cycles. For me it was a great cardio workout. Weights and reps can of course be adjusted to suit ones personal conditioning. Doubt I'll ever be able to push that hard again given my condition, but for anyone in decent shape it's a nice cardio. Much less boring than the elliptical or tread mill.