The plane is simply connected, but the plane without the origin is not. Draw a circle around the origin, and we cannot shrink it down to a point. No matter how we slide the circle about, it always crashes into the "hole" at the origin. Similarly, 3 space without the x axis is not simply connected.
What about 3 space without the origin? This is simply connected, by the above definition. A circle around the origin can always move out of plane, then shrink to a point. However, there are times when the definition changes with the dimension. It all depends on context. If spheres must shrink continuously to points, then 3 space without the origin is no longer simply connected, as the unit sphere about the origin cannot shrink to a point.
Unless otherwise stated, simply connected refers to paths and circles, regardless of the dimension of the containing space. This is the case when dealing with conservative fields in physics.
None of this is terribly rigorous. We will explore this topic further when homotopies have been introduced as part of algebraic topology.