What they draw is not what we would typically draw in a physics class. Your instincts are correct. The velocity vector is straight ahead (no matter how the wheel is turned), and the force of the wheels is pulling it in the directly of the center of the curve.
Unfortunately, this is not natural for many people. Many people's intuition does not jive with this. Instead of trying to give a physics lesson, they're trying to give a safety lesson. What they draw as "the direction of inertia" is the direction of the velocity of the car if the car was not turning. They then draw the "direction of steering" as the current velocity vector.
There is no useful physics interpretation of this. Nobody would use this to, say, calculate the forces on the struts to make sure they can handle this particular radius turn at a given speed. They would do that in the way you are thinking. However, when you're driving, the "direction of steering" really does feel like the direction you are steering in, and the idea of this "direction of inertia" is close enough to the feeling of the "fictitious" centrifugal force (fictitious because it isn't a real force, but rather a re-interpretation of other forces in a rotating frame). The idea is to get people to have an intuitive grasp of what can cause them to leave the road if they speed.
When it comes to safety and other place, physical accuracy gives way to intuitive readability. The key here is that, if people have to think about it (i.e. physics lesson), they will not be able to apply it in the half second where it matters on the road.