Exterior covariant derivative

Schuller_2013 page 197
Olver_2014 page 31
Olver_1995 page 123-124

Idea: when we have a decomposition of the cotangent bundle in horizontal and vertical subspaces, we can use the usual exterior derivative operator d and then project the result on the horizontal subspace. This leads to a new operator dH.

I think that it has to do with the typical situation of a function with parameters (like in high school). Consider f(x)=ax3+x23x+1. We have variables x and a, but they are not of the same "status". Students ask why we derive x but not a...

Even in function of several variables

f(x,y)=ax2+y2

they don't have the same "status": x and y are more "variables" and a is more fixed.
I think that the exterior covariant derivative is an "external device" that we introduce in order to specify this distinction. Indeed I guess that the connection itself has this goal...