Am I riding my bike wrong?

No...your feet should not be able to touch the ground while stopped & still sitting on the saddle or seat (whichever you prefer). Your knees hurt because they're not getting the proper extension. Here's an easy way to check... With the balls of your feet centered on the pedals, on the down-stroke there should be a slight bend in the knee; maybe 10 to 15 degrees. Method #2; Don’t ride like this...just a quick check. With the heel of your foot centered on a pedal, on the farthest point of the down-stroke your leg should be straight.

Starting and Stopping from the late cycling Guru, Sheldon Brown. https://www.sheldonbrown.com/starting.html   When coming to a stop, you slide off the saddle & straddle the bike. If this 13-year-old girl can do it, you can too. https://youtu.be/9XVWeZgcskU

If your seat is too low it will cause knee injury. That website is wrong. 

There ARE bikes where you can reach the ground flat-footed from the saddle. Usually called semi-Recumbents or pedal-forward bikes. 

But on bikes with more common geometries, you’re not expected to be able reach the ground flat-footed from the saddle. RaleighBob gives good advice on saddle height adjustment.