There is difference between an open boat and a closed boat as the air inside a closed boat is contained inside while the air "on" an open boat is always moving.
No there is not. The only part of the air that counts is the air below the waterline. If you make a sphere, it will displace the exact amount of water as if you make an open boat. You could make a sphere with a hole in the top if you want, it will float the exact same way.
If the wooden boat is left in water for too long it still sinks.
This is clearly wrong. As long as the boat doesn't have a leak in it, it will float.
How can it include the air's volume and density as the air is always moving?
The air is not important, the displaced volume is. You you pumped all the air out of the boat, it would float just as well (actually slightly better because you have lowered the mass of the boat). The key thing is that the shape you use is pushing water out of the way and that displaced volume reduces the effective density of your boat. The key factors are the weight of the boat, compared to the weight of the displaced water. By shaping your material to displace more water, you can support a heavier boat. It doesn't matter if the boat is open to air or not.
But if you said so you are saying that the textbook had taught us wrongly?
No, I'm saying you have not understood what the book is trying to teach you. When you make a boat from steel, you do not reduce the actual density of steel by reshaping it. I'm sure your book does not say that.