With respect to being parallel, a connection on a vector bundle is not directly a vector field *per se*, but it lifts every vector field X on the base to a vector field on the bundle. Being parallel with respect to the connection is then equivalent to being parallel with respect to each of these , which is indeed the same as being closed with respect to parallel transport.

First of all, there are again some mistakes with numbering. The numbers 1, 2, 3, 13 has been assigned twice.

I have also a question about the definition of “preservation of K by parallel transport”, How the covariant differentiation with respect to X is a vector field?

If we think of K as a plane distribution, then we can interprete this as closedness of K under covariant differentiation. But, Is this equivalent to closedness of K under parallel transportion? Or does this implies it?