Self-Shadowing problems on MacBook Pro

OK, I just found the problem to be an old Vertex-Lit shader... If I remove this one shader file from the project, the problems are gone. I'll try to look deeper into it if I have the time.

Thanks to all!

Edit for Joseph:

Here is the shader's code. I don't know where we got it from initially. Most likely a left-over from a previous version of the built-in shaders package Unity provides on their website, or from the community wiki. Maybe you can identify the code that caused the problem.

 Shader "VertexLit" {
 Properties {
     _Color ("Main Color", Color) = (1,1,1,1)
     _SpecColor ("Spec Color", Color) = (1,1,1,1)
     _Emission ("Emmisive Color", Color) = (0,0,0,0)
     _Shininess ("Shininess", Range (0.01, 1)) = 0.7
     _MainTex ("Base (RGB)", 2D) = "white" {}
 }
 
 SubShader {
     // Normal rendering pass
     Pass {
         Material {
             Diffuse [_Color]
             Ambient [_Color]
             Shininess [_Shininess]
             Specular [_SpecColor]
             Emission [_Emission]
         } 
         Lighting On
         SeparateSpecular On
         SetTexture [_MainTex] {
             Combine texture * primary DOUBLE, texture * primary
         } 
     }
     
     // Pass to render object as a shadow caster
     Pass {
         Name "ShadowCaster"
         Tags { "LightMode" = "ShadowCaster" }
         
         Fog {Mode Off}
         ZWrite On ZTest Less Cull Off
         //Offset [_ShadowBias], [_ShadowBiasSlope]
 
 CGPROGRAM
 #pragma vertex vert
 #pragma fragment frag
 #pragma multi_compile SHADOWS_NATIVE SHADOWS_CUBE
 #pragma fragmentoption ARB_precision_hint_fastest
 #include "UnityCG.cginc"
 
 struct v2f { 
     V2F_SHADOW_CASTER;
 };
 
 v2f vert( appdata_base v )
 {
     v2f o;
     TRANSFER_SHADOW_CASTER(o)
     return o;
 }
 
 float4 frag( v2f i ) : COLOR
 {
     SHADOW_CASTER_FRAGMENT(i)
 }
 ENDCG
 
     }
     
     // Pass to render object as a shadow collector
     Pass {
         Name "ShadowCollector"
         Tags { "LightMode" = "ShadowCollector" }
         
         Fog {Mode Off}
         ZWrite On ZTest Less
 
 CGPROGRAM
 // Upgrade NOTE: excluded shader from Xbox360; has structs without semantics (struct appdata members vertex)
 #pragma exclude_renderers xbox360
 #pragma vertex vert
 #pragma fragment frag
 #pragma fragmentoption ARB_precision_hint_fastest
 
 #define SHADOW_COLLECTOR_PASS
 #include "UnityCG.cginc"
 
 struct appdata {
     float4 vertex;
 };
 
 struct v2f {
     V2F_SHADOW_COLLECTOR;
 };
 
 v2f vert (appdata v)
 {
     v2f o;
     TRANSFER_SHADOW_COLLECTOR(o)
     return o;
 }
 
 half4 frag (v2f i) : COLOR
 {
     SHADOW_COLLECTOR_FRAGMENT(i)
 }
 ENDCG
 
     }
 }
 }