I then imported it to Illustrator and now I am trying to colour it exactly as the logo below. Any help?
You have to understand how vector graphics work; it's not the same as painting on a piece of paper, where every brush stroke covers what you are painting over.
You also have to shift your thinking a little away from filling shapes: since you can create independent objects, the outline can be a different object than the fill color object. See where I'm going with this?
You already have an outline object. It has open shapes, and although you can still fill it with color (it doesn't matter how at this point) it will follow the shape of the outline. Instead, create a different shape, fill it with the color you want and send it to the back of your outline object. This way you can get the same results from your first example.