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The first parametrization is the parametrization of a solid cone. To plot it, you can plot the boundary, which is where $s=\theta$ or $u=h$:

theta=pi/6
h = 60
var('s,t,u')
parametric_plot3d((u*tan(theta)*cos(t),u*tan(theta)*sin(t),u),(u,0,h),(t,0,2*pi) ) + parametric_plot3d((h*tan(s)*cos(t),h*tan(s)*sin(t),h),(s,0,theta),(t,0,2*pi) )

The first parametrization is the parametrization of a solid cone. cone.

To plot it, you can plot the boundary, which is where $s=\theta$ or $u=h$:

theta=pi/6
sage: h = 60
var('s,t,u')
parametric_plot3d((u*tan(theta)*cos(t),u*tan(theta)*sin(t),u),(u,0,h),(t,0,2*pi) ) 20
sage: theta = pi/3
sage: side = parametric_plot3d((lambda t, u: u*tan(theta)*cos(t),
....:                           lambda t, u: u*tan(theta)*sin(t),
....:                           lambda t, u: u), (0, 2*pi), (0, h))
sage: base = parametric_plot3d((lambda s, t: h*tan(s)*cos(t),
....:                           lambda s, t: h*tan(s)*sin(t),
....:                           lambda s, t: h), (0, theta), (0, 2*pi))
sage: side + parametric_plot3d((h*tan(s)*cos(t),h*tan(s)*sin(t),h),(s,0,theta),(t,0,2*pi) )
base