I am wondering what are the names of the specific names of the lines used when a program such as Matlab or Mathematica plots a surface in 3D. They are lines which are the intersection between planes parallel to the axis and the surface itself.
As opposed to say contour lines or
And what are the lines usually presented with sphere?
The word you're looking for is "trace." In usage one typically specifies the direction to take the trace, e.g. "This family of curves is generated by taking traces in the $xy$-direction.
Depending on the context, one might also use "gradient curves," "contours," or "level curves/sets." However, these are usually reserved for more specialized circumstances.