Plotting ErrorBars with a different style

You do not have to re-implement anything, just use a little chicanery:

ErrorListPlot[
 data,
 PlotStyle -> Directive[Thick, PointSize -> 0.015],
 ImageSize -> Large,
 PlotMarkers -> Automatic,
 ErrorBarFunction -> (ErrorBarPlots`Private`ebarfun[##] /. l_Line :> {[email protected], l} &)
 ]

Here is an alternative you might like based on an answer I got to a previous question of mine: Plotting Error Bars on a Log Scale

ClearAll[ePlot, ePlotFunc, plusMinusMean];
plusMinusMean[a_, b_] := {a + b, a - b, a};
ePlot[plotFun_, dataX_, plusMinList_, color_] := Block[{f},
  f[y_] := Transpose[{dataX, y}];
  plotFun[{
    f[plusMinList[[All, 1]]], f[plusMinList[[All, 2]]], 
    f[plusMinList[[All, 3]]]}, Filling -> {1 -> {2}}, 
   Joined -> {True, True, True}, 
   PlotStyle -> {Opacity[0], Opacity[0], Darker@color}, 
   PlotMarkers -> {Graphics@{Disk[]}, 0.03}, 
   FillingStyle -> Directive[Opacity[0.2], color], Frame -> True, 
   Axes -> False]]


ePlotFunc[data_, color_] := 
 Module[{dataY, dataX, errorY, plusMinList}, 
  dataY = data[[All, 2]];
  dataX = data[[All, 1]];
  errorY = data[[All, 3]];
  plusMinList = Thread[plusMinusMean[dataY, errorY]];
  ePlot[ListPlot, dataX, plusMinList, color]]

colors = {Hue[0.67, 0.6, 0.6], Hue[0.85, 0.6, 0.6]};

data = Table[{x, f[x], RandomReal[]}, {f, {Exp[2 #] &, Exp}}, {x, -5, 
    2, 0.2}];

Show@MapThread[ePlotFunc, {data, colors}]

I hope you might be able to adapt this to your needs, see another example:

Points are stylled with PlotStyle, Lines with ReplaceAll:

ErrorListPlot[data,
  PlotStyle -> Directive[Thick, PointSize -> 0.01], ImageSize -> Large
  ] /. x_Line :> {[email protected], x}   

(it was Sequence[[email protected],x] before rcollyer comment)

It is not allways so short, in case of more objects in Plot/Show etc. one has to specify pattern more carefully.