If you will look right at the bottom corner of your post, you will see a reply button. There is one on my post too. To get what you see, where the person sees you replied directly to them, you must click that one. This post I clicked the blue reply button here
This button allows you to reply to the thread without it being to a particular person. Great if you want to quote and reply to multiple people.
So basically, if you want to respond to a specific person, click the reply button within their post rather than the blue reply button. That also makes that person's icon appear so the rest of us know it's a direct reply to someone else.