Surface and Plane : Are they different in 3D vector?

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Hopefully it's a question.

In 2D geometry the graph of an equation involving x and y is a curve in the plane. In three-dimensional geometry an equation in x, y, and z represents a surface in space.

I couldn't understand about the Surface, here's a picture I modified through ms. paint:

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There's a thing that is even confusing me :

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Is this a plane ? The thing is its width is over increasing along the y-axis while it should be y=0 Then what would I call it in my B-picture above?