This isn't stressed near enough in MSF courses. The instructor [in my case] kept yelling, "press & roll, press & roll" during one of our exercises. As one reader astutely pointed out in an earlier response, at low speeds [as in an MSF class], the bike tracks the way the yoke is turned, hence no counter steer.
I first experienced this phenomena during my second or third time out on the road after getting my license. It's a tough subject to attack as the last thing you'd want is to introduce too much speed in MSF where there are riders [like myself] who have never been on a bike, but somehow we need to better prepare riders in this area.
It is when I teach, but each coach has their unique way of explaining it. It's first introduced in class, but it's kind of difficult explaining it until you're actually on the motorcycle doing it.
The best way I get my riders to understand how countersteering works is that they first have to establish speed, greater than 10 mph. You have to get both wheels of the bike to roll fast enough to do it. If you don't, then you will be steering the handle bars like you would on a tricycle.
When they actually sit on the bike, and have them practice pressing against the handlebars while they are just sitting and simulating a riding position, I often will select a student to put up both hands and place them forward against the palm of my hands and ask them to press right, then left. Everyone then starts to get the idea. Press right, go right, press left, go left. Once again, only after you have acheived a fast enough speed.