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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!

You can use MeshFunctions + Mesh + MeshStyle to mark the intersection of the plane with the surface:

pp = ParametricPlot3D[Surface[u, v], u, -Pi, Pi, v, -Pi, Pi, 
 PlotStyle -> Opacity[0.3], 
 PlotPoints -> 200, 
 MeshFunctions -> #3 &, 
 Mesh -> 10^-6, Red];

 Show[pp, Graphics3D[Blue, Opacity[0.4], 
 InfinitePlane[0, 0, 0, 1, 0, 0, 0, 1, 0]]]

Extract the line primitives from pp and remove the last column of coordinates to get 2D lines to be used with Graphics:

Graphics[Red, Thick, Cases[Normal[pp], Line[x_] :> Line[x[[All, ;; 2]]], All], 
 Frame -> True]

Use (1) MeshFunctions -> #2 & in ParametricPlot3D, (2) InfinitePlane[0, 0, 0, 1, 0, 0, 0,0,1] in Graphics3D and (3) Line[x[[All, 1,3]]] in Cases to get

And MeshFunctions -> #2-# &, InfinitePlane[0, 0, 0, 1, 1, 0, 0,0,1], ] :> Line[x[[All, 1,3]]] to get

Here my solutionapproach

First define the plane e and n

n = 0, 0, 1; (*normalvector of plane*)
e = 0, 0, 0; (*point of plane*) 

Condition of intersection is Surface[u, v] - e).n == 0 which can be visualized with

ContourPlot[(Surface[u, v] - e).n == 0, u, 0, 2 Pi, v, -1.1 Pi ,1.1 Pi , FrameLabel -> u, v,PlotRange -> 0, 2 Pi, -1.1 Pi, 1.1 Pi]

Obviously the intersection is defined by 0<u<2Pi ,v [Element] 0, Pi (which could be evaluated by NSolve[(Surface[u, v] - e).n == 0,0 <= u <= 2 Pi, -Pi <= v <= Pi, u, v, Reals]):

ParametricPlot3D[Surface[u, 0], Surface[u, Pi], u, 0, 2 Pi]

It's not necessary to introduce new coordinates x,y.

2nd example:

n=0,1,0;
sol = NSolve[(Surface[u, v] - e).n == 0,0 <= u <= 2 Pi, -Pi < v < Pi, u, v, Reals] 
ParametricPlot3D[Surface[u, v] /. sol, u, 0, 2 Pi]