How to plot a crossection of a ParametricPlot3D?

I’m plotting a surface using ParametricPlot3D. Ideally I would like to plot an intersection of the surface with a plane just as seen in another question (Cross Sections or slices of 3d figures – Mathematica). However in my case the parameters u and v are not equivalent to the spatial coordinates x,y. This is why i tried RegionPlot in place of CountourPlot to plot all pairs (x,y) that intersect the surface for a given value of z. This is not working somehow and I’m lost trying to look for alternatives.

Surface[u_, v_] = {Cos[u], Sin[u] + Cos[v], Sin[v]};  Row[{Show[{ParametricPlot3D[      Surface[u, v], {u, 0, 2Pi}, {v, -Pi, Pi},       PlotStyle -> Opacity[ 0.2]],      Graphics3D[{Blue, Opacity[0.4],        InfinitePlane[{0, 0, 0}, {{1, 0, 0}, {0, 1, 0}}]}]},     ImageSize -> 300],    RegionPlot[    Exists[{u, v}, Surface[u, v] == {x, y, 0}], {x, -1, 1}, {y, -1, 1},     ImageSize -> 300]}] 

In reality my surface is a bit more complicated so I chose a simple one for posing this question. Also ideally I would like to be able to change around the normal vector of the plane later, checking out any angle of the crosssection I like, so please keep that in mind when thinking about the question. Thank you very much for any engagement in my problem!