TikZ: How to draw a pattern at the border of a tikz path

NEW ANSWER BASED ON YOUR OWN ANSWER: This avoids the white filling. UPDATE: One single style does the job. (I also did the blue contour.)

documentclass{minimal}
usepackage{tikz}
usetikzlibrary{patterns}
usetikzlibrary{decorations,backgrounds}

newcounter{tmp}

%tikzstyle{D}=[draw=blue,line width=1pt,preaction={clip, postaction={draw=blue,opacity=0.5,line width=12pt}}]
%<- note that tikzstyle is deprecated

tikzset{D/.style={
preaction={draw=blue,line width=1pt},
preaction={decoration={contour lineto closed, contour distance=6pt},
decorate,
},
postaction={
insert path={%
pgfextra{%
pgfinterruptpath
begin{scope}[opacity=0.5, transparency group]
path[fill=blue,even odd rule] 
mySecondList myList 
;
end{scope}
endpgfinterruptpath}
}},
}}



tikzset{EDR/.style={
preaction={draw=red,line width=1pt},
preaction={decoration={contour lineto closed, contour distance=6pt},
decorate,
},
postaction={
insert path={%
pgfextra{%
pgfinterruptpath
path[pattern=north west lines, pattern color=red,even odd rule] 
mySecondList myList 
;
endpgfinterruptpath}
}},
}}

makeatletter
defpgfdecoratedcontourdistance{0pt}
pgfset{
  decoration/contour distance/.code=%
    pgfmathsetlengthmacropgfdecoratedcontourdistance{#1}}
pgfdeclaredecoration{contour lineto closed}{start}{%
  state{start}[
    next state=draw,
    width=0pt,
    persistent precomputation=letpgf@decorate@firstsegmentanglepgfdecoratedangle]{%
    pgfextra{xdefmyList{}xdefmySecondList{}}
    pgfextra{setcounter{tmp}{0}}
    pgfpathmoveto{pgfpointlineattime{.5}
      {pgfqpoint{0pt}{pgfdecoratedcontourdistance}}
      {pgfqpoint{pgfdecoratedinputsegmentlength}{pgfdecoratedcontourdistance}}}%
  }%
  state{draw}[next state=draw, width=pgfdecoratedinputsegmentlength]{%
    ifpgf@decorate@is@closepath@%
      pgfmathsetmacropgfdecoratedangletonextinputsegment{%
        -pgfdecoratedangle+pgf@decorate@firstsegmentangle}%
    fi
    pgfmathsetlengthmacropgf@decoration@contour@shorten{%
      -pgfdecoratedcontourdistance*cot(-pgfdecoratedangletonextinputsegment/2+90)}%
    pgfpathlineto
      {pgfpoint{pgfdecoratedinputsegmentlength+pgf@decoration@contour@shorten}
      {pgfdecoratedcontourdistance}}%
    stepcounter{tmp}
    pgfcoordinate{muemmelthetmp}{pgfpoint{pgfdecoratedinputsegmentlength+pgf@decoration@contour@shorten}
      {pgfdecoratedcontourdistance}}
    pgfcoordinate{feepthetmp}{pgfpoint{pgfdecoratedinputsegmentlength}{0pt}}      
    pgfextra{xdefmyList{myList (muemmelthetmp) -- }%
        xdefmySecondList{mySecondList (feepthetmp) -- }}
    ifpgf@decorate@is@closepath@%
      pgfpathclose
      pgfextra{xdefmyList{myList cycle}% 
      xdefmySecondList{mySecondList cycle}}
    fi
  }%
  state{final}{pgfextra{%typeout{myList,mySecondList}
  }}%
}
makeatother
tikzset{
  contour/.style={
    decoration={
      name=contour lineto closed,
      contour distance=#1
    },
    decorate}}
begin{document}

begin{tikzpicture}
draw(0,0) -- ({sqrt(8)},4) node[midway,sloped,above]{That's just a test!};

path[EDR]
(1,0) -- (4.5,1.5) -- (4,4) -- (1,3.5) -- (3.5,2) -- cycle;

path[D] (7,0) -- (9.5,2.5) -- (11,4) -- (9,3.5) -- (8,2) -- cycle;


end{tikzpicture}

end{document}

ORIGINAL ANSWER: Some very similar question has been answered here. Using the code written there allowed me to write a command DrawBorder, which I believe does what you want. Note, however, that the present version works for polygons only. (EDIT: Added the BCD style, cleaned up the code and added explanations.)

documentclass[tikz,border=3pt]{standalone}
usetikzlibrary{patterns,decorations,calc}
%
defcontourwidth{12pt} 
% Notice that this width enters at two places
% first it defines the widths of the nodes created by tikzsegment
% but it also defines the overshoot, required when an angle is larger than 
% 180 degrees
newcommand{tikzsegment}[3][]{ % from https://tex.stackexchange.com/a/192824/121799
path let
      p1=($#3-#2$),
      n1={veclen(p1)+1.75*contourwidth}
 in #2 -- #3 
 node[minimum width=n1, 
         inner sep=0pt, 
         pos=0.5,sloped,rectangle,
         fill=white]{} 
    node[minimum width=n1, 
         inner sep=0pt, 
         pos=0.5,sloped,rectangle,
         #1] 
     (line){};
}
newcommand{DrawBorderA}[2][]{
begin{scope}
foreach point [count=n] in {#2} {
ifnumn=1
xdefClipList{point --}
else
xdefClipList{ClipList point --}
fi
node (prev) at point {};
}
xdefClipList{ClipList cycle;}%typeout{ClipList}
clip ClipList
foreach point in {#2} {
node (new) at point {};%typeout{point}
tikzsegment[pattern=north west lines, pattern color=red,
               minimum height={2*contourwidth}]
              {(prev.center)}{(new.center)}    
node (prev) at point {};            
}
draw[red,line width=2pt] ClipList                    
end{scope}
}

newcommand{DrawBorderBCD}[2][]{
begin{scope}[opacity=0.5, transparency group]
foreach point [count=n] in {#2} {
ifnumn=1
xdefClipList{point --}
else
xdefClipList{ClipList point --}
fi
node (prev) at point {};
}
xdefClipList{ClipList cycle;}%typeout{ClipList}
clip ClipList
foreach point in {#2} {
node (new) at point {};%typeout{point}
tikzsegment[fill=blue,
               minimum height={2*contourwidth}]
              {(prev.center)}{(new.center)}
node (prev) at point {};            
}             
end{scope}
draw[blue,opacity=1,line width=1pt] ClipList            
s}

% tikzstyle{EDR}=[draw=red,line width=1pt,preaction={clip, postaction={pattern=north west lines, pattern color=red}}]
% tikzstyle{D}=[draw=blue,line width=1pt,preaction={clip, postaction={draw=blue,opacity=0.5,line width=12pt}}]
begin{document}

begin{tikzpicture}

DrawBorderA{(1,0),(4.5,1.5),(4,4),(1,3.5),(2.5,2)}

begin{scope}[xshift=5cm]
DrawBorderBCD{(1,0),(4.5,1.5),(4,4),(1,3.5),(2.6,2)}
end{scope}

end{tikzpicture}
end{document}

DISCLAIMER: It does not yet work with arbitrarily crazy angles (much larger than 270 degrees). Dealing with those will either require brute force, i.e. some fair amount of work, or some clever idea. I plan to revisit this task once I know that this is the way to go.