First off... if it's gyroscopic precession that's responsible for everything, then why does different bikes with different head-tube angles and fork-rakes steer differently with the same wheel?
When you ride no hands, you'll notice that the steering waggles back and forth in response to your pedaling rhythm. Pay attention to what happens to the steering each time you push down on each leg. When you push down on the left leg, it leans the bike left and you'll see that the steering aims right. Then when the right leg pushes down, it leans the bike to the right and the steering aims left. When you steer around a corner no hands, your weight-shift pulls the bike to one side and makes the steering turn to the opposite side. You lean left and the steering aims right.
A REAL clear way to illustrate countersteering is to do laps around a parking-lot ONE-HANDED. Take a simple large 90-degree right-hand corner (over 10mph) with your right-hand on the bars. Which way do you have to move your hand in order to make it turn right? You'll have to PUSH with your right hand to make it steer LEFT in order to turn right. Similary, if you had your left hand on the bars, you'll have to PULL back with left-hand to steer LEFT to make bike go right around the corner. I teach this exercise to all 1st-year members of our collegiate team every single year.
A lot of people can't ride smoothly and hold their line around corners because they don't understand countersteering and they end up sending conflicting signals with both their hands. One hand's pushing one way, the other pushing the other way, their brain's saying steer into the corner, yet their butt feels something opposite. The usual result is they stay upright and fly off the outside of the corner. The one-handed exercise really clear up which way you steer in order to make a turn.
As for the OP's question of "squirrely", the opposite would be:
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stable: doesn't respond very quickly
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heavy: takes a lot of force on the bars to make the bike turn
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steers like a FAT PIG: if you've ever tried riding a hog, you'll know what this means
And those mentioning it's the rider are1 exactly right. Any and every single bike will ride itself in a perfectly straight line. Test it yourself, get up to 10-15mph and hop off the back of the bike without making it twitch, it'll ride in a perfectly stable straight line. You see it in motorcycle-racing all the time, some guy gives it too much throttle coming out of a corner and high-sides. The bike throws him off, then it recovers and ghost-rides itself in a straight line. Until it hits bumps and causes the steering to turn and it'll start carving a curve into tighter and tighter spirals until it falls over. It's like Newton's laws of motion, the bike's gonna go in a straight line until some external force makes it change directions.
In most cases, it's the rider and the rider's response to road conditions that causes twitchiness. Hanging onto the bars too stiffly will cause darting behavior when you hit bumps. Bending your elbows, leaning over the bars and absorbing the road shocks will make you more stable. Personally, I like a bike that's as twitchy as possible, it'll steer telepathically. I miss the old Trek 770 crit bike, 74.5-degree head angle, 39.5cm chainstays, 37.5" wheelbase... ahhh, you don't actually ride ON that bike, you ride IN it, like it's a part of you.
BTW - once you've tipped it over to the correct lean-angle, you undo the countersteering. At this point, camber-roll/thrust makes the tyre carve a curved path on the road.