Quote by Gadhav
These terms are confusing since some books insist that Contravariant component is parallel to axes and Covariant component is perpendicular.

Quote by Fredrik
I don't think I've seen this claim, and I don't understand it.

He's probably talking about the ability to identify forms and tangent vectors in the presence of a metric (lowering and raising indices). Something like
http://www.mathpages.com/rr/s502/502.htm "As can be seen, the jth contravariant component consists of the projection of P onto the jth axis parallel to the other axis, whereas the jth covariant component consists of the projection of P into the jth axis perpendicular to that axis."
That's a different approach from what you've taken where you define forms and vectors first without a metric. Then only with a metric are we allowed to identify forms and vectors. In this approach, the metric always exists, which is an assumption of classical GR.