Lemon-Limify a String
Matlab, 140 136 128 124 bytes
Basically first starts with the middle section, and then prepends/appends the shortened/modified versions step by step.
a=input('');v=ones(nnz(a)*2-1,1)*a;v(2:2:end,:)=0;b=v;for k=a;v=v(2:end,1:end-1);v(end+1,:)=0;b=[v,b,v,''];end;b(~flip(b))=0

Thanks for 8 bytes @LuisMendo!
E.g. for MATLAB we get:
               MATLAB               
          MATLA      MATLA          
      MATL     MATLAB     MATL      
   MAT    MATLA      MATLA    MAT   
 MA   MATL     MATLAB     MATL   MA 
M  MAT    MATLA      MATLA    MAT  M
 MA   MATL     MATLAB     MATL   MA 
   MAT    MATLA      MATLA    MAT   
      MATL     MATLAB     MATL      
          MATLA      MATLA          
               MATLAB