Point 2 is a simplification that borders on wrong: you don't "wrap around if it's too big" - the point operation will always cross the curve (except for the zero point)!
You "wrap around" because you're using a finite field. As far as I understand, the finite field of the scalars allows you to have an inverse for the multiplication, and the finite field of the points is there to make the calculation easier.
You "wrap around" because you're using a finite field. As far as I understand, the finite field of the scalars allows you to have an inverse for the multiplication, and the finite field of the points is there to make the calculation easier.