You only want to show the contour for the solution f == 0 which can be done with ContourPlot. For the Plot3D all you have to do is set the proper plot range (and remove the boundary and clipping styles). Or you can use the parametric equation and use parametric plot.
ContourPlot3D[x^2 - y^2 z == 0, {x, -1, 1}, {y, -1, 1}, {z, 0, 1}]
ContourPlot3D[x^2 - y^2 z, {x, -1, 1}, {y, -1, 1}, {z, 0, 1},
Contours -> {0}]
ParametricPlot3D[{u v, u, v^2}, {u, -1, 1}, {v, -1, 1}]
Plot3D[x^2/y^2, {x, -1, 1}, {y, -1, 1}, PlotRange -> {0, 1},
ClippingStyle -> None, BoundaryStyle -> None]
These all work