If you reuse your sim card and simply place the old sim in a mew phone, there should not be charge. I do this and I do not get charged. However, if you get a new sim, you will pay. That has been my experience. The problem is most people do reuse the sim. If you go to apple they sell you a phone with a sim already in there, they will not switch the sim over and the same is true at most telco stores.
New SIMs are free from AT&T. All too often after swapping SIMs, I have had to go into the AT&T store to get a new SIM. These micro versions seem especially fragile. No charge for anything. They scan, insert, turn on and hand back to me.
I've even changed due to size restrictions (mini to micro...) and there has not been a charge.
I think the article has it incorrect or is missing a portion.