The rotation error between two frames can be viewed in two ways:
The orientation of one frame as seen from the other, calculated by multiplying the inverse of the observing frame by the observed frame. For frames $E$ and $H$, this error in your notation would be
$$
\tag{1}
{R_{1}}^{E}_{H} = (R^{W}_{E})^{-1}R^{W}_{H} = R^{E}_{W}R^{W}_{H}
$$
for $H$ as seen from $E$, or
$$
\tag{2}
{R_{1}}^{H}_{E} = (R^{W}_{H})^{-1}R^{W}_{E} = R^{H}_{W}R^{W}_{E}
$$
for $E$ as seen from $H$. These matrices encode the rotations around local axes (the axes of the superscript frame) that map the superscript frame to the subscript frame,
$$
\tag{3}
R^{W}_{E}{R_{1}}^{E}_{H} = R^{W}_{H}
$$
and
$$
\tag{4}
R^{W}_{H}{R_{1}}^{H}_{E} = R^{W}_{E}
$$
The orientation that would be reached by starting at one frame and moving by the inverse of the second frame's orientation relative to the world,
$$
\tag{5}
{R_{2}}^{E}_{H} = R_{H}^{W} (R_{E}^{W})^{-1} = R_{H}^{W} R_{W}^{E}
$$
or
$$
\tag{6}
{R_{2}}^{H}_{E} = R_{E}^{W} (R_{H}^{W})^{-1} = R_{E}^{W} R_{W}^{H}
$$
These matrices encode the rotations around the world axes that map the frames to each other,
$$
\tag{7}
{R_{2}}^{E}_{H} R^{W}_{E} = R^{W}_{H}
$$
and
$$
\tag{8}
{R_{2}}^{H}_{E}R^{W}_{H} = R^{W}_{E}
$$
The fundamental source of there being two interpretations of rotational error is that rotations are not commutative, and so our intuition that the differences $A + (-B)$ and $(-B) + A$ should be the same does not hold in this case.
The equations have a left-right duality. When we take the difference by putting the inverse rotation on the left (globally rotating the second frame by the inverse of the first), we find the (local) rotation error that maps between the frames when placed to the right of the starting frame, and when we put the inverse rotation on the right (locally rotating the first frame by the inverse of the second), we find the (global) rotation error that maps between the frames when placed to the left of the starting frame.