It's wildly unintuitive, so don't feel bad. When I was first learning to ride a motorcycle with the MSF, I damn neared dumped the practice bike a few times trying to manhandle it. You'll really quickly internalize push-left-go-left. And it turns out to work great on regular bikes, too. (But it's terrifyingly more responsive - I actually can't relax pedaling a bicycle because of the fear of how easy it is to flip the things and capsize them from countersteering.)
EDIT: There's a neat heuristic that was brought up in the course: the center of mass of the bike is above the bottom of the wheels, and a turning bike must lean into the turn (weird road geometries excepted), therefore to turn the stuff under the center of mass must counter the top leaning in. Thus to go left, you push left, causing the front wheel to pull out from under the bike right-wards, which starts to capsize the bike leftwards! Like I said before, a bicycle reacts really really hard to this, but a hefty motorcyle has a far lower center of mass and leans into it more gracefully.
> And it turns out to work great on regular bikes, too
Well yes, seeing as that's how you steer a bicycle. As a keen cyclist it annoys me when people (especially coaches) start harping on about countersteering being an additional technique you should learn, when it's what everyone has been doing intuitively since they first learned to ride a bike.
We learn counter-steering subconsciously, right? It's intuitive in the sense that it's muscle memory, but not in the sense that it's conscious and obvious why. A lot of people who've ridden bicycles initially refuse to believe counter-steering exists, I think that demonstrates that it's not completely intuitive. If the counter-steering part is obvious to you, then you're probably ahead of the curve, and you can certainly ignore when coaches harp about it.
Counter-steering is said to be important to learn consciously because it's unintuitive, and people have crashed and died on motorcycles in panic situations by turning their handlebars toward their escape route only to have the bike go the opposite direction. Or at least I heard a rumor about that in my MSF course.
I do think it's valuable to know this fact consciously and not rely on body intuition, but I'm also sure it's much more important for motorcycling than for bicycling.
Don't take it personally - I was always just a terrible rider when I was young; I meant that in a mildly sarcastic way. I never really figured out how to properly steer until I got a motorcycle, and that just about ruined my feeling of force in the other way. Put me on a bike and in 15 feet I'll accidentally steer so hard you'd think I was dodging semis / saw a hundred dollar bill on the pavement. (I assume as a kid I just always manhandled the bike so hard I never noticed. Always standing on the pedals and pumping them like I was running, wobbling the bike's center of gravity the whole way.)
EDIT: There's a neat heuristic that was brought up in the course: the center of mass of the bike is above the bottom of the wheels, and a turning bike must lean into the turn (weird road geometries excepted), therefore to turn the stuff under the center of mass must counter the top leaning in. Thus to go left, you push left, causing the front wheel to pull out from under the bike right-wards, which starts to capsize the bike leftwards! Like I said before, a bicycle reacts really really hard to this, but a hefty motorcyle has a far lower center of mass and leans into it more gracefully.