Shader to make a quad work like a canvas

I am trying to learn a bit about shaders.

What I am trying to do right now, is to set up a Vertex Shader in such a way that I can use a Quad as a canvas to draw animations on using only the fragment shader, which is for instance what is used on https://www.shadertoy.com

Right now my Shader looks like the following:

 Shader "Custom/FragmentShaderTest" {
     Properties {
         _Color("Color", Color) = (1.0,1.0,1.0,1.0)
     }
     SubShader{ 
         
         Pass{
         CGPROGRAM
             #pragma vertex vert
             #pragma fragment frag
             
             //user defined variables
             uniform float4 _Color;
             
             struct vertexInput{ 
                 float4 vertex: POSITION;
             };
             
             struct vertexOutput{ 
                 float4 pos: SV_POSITION;
                 float4 posWorld: TEXCOORD0;
             };
             
             vertexOutput vert(vertexInput v){
                 vertexOutput o;
                 
                 o.posWorld=mul(UNITY_MATRIX_MVP, v.vertex);
                 
                 o.pos = mul (UNITY_MATRIX_MVP, v.vertex);
                 
                 
                 return o;
             }
             
             float4 frag(vertexOutput i):COLOR
             { 
                 float2 uv = i.posWorld.xy;
                 uv= 2.*uv-1.;
                 float dist = distance(uv, float2(0.5,0.5));
                 float4 col;
                 
                 col = float4(uv.x,uv.y,1.0,0.5);
                 col *=dist;
                 
                 return col;
             }
             
             
             
         ENDCG
         }
     }
     //Fallback "Diffuse"
 }

However, the position that is given to the fragment shader from the vertex shader is not the local position on the plane; it changes when I move my camera or rotate the object.

How do I pass the position along in the correct way?