The notation for tensors is like the plumbing in a very old Vermont farmhouse. It may once have been intentionally designed, but after that it just evolved. As an example, it seems that depending on taste in notational systems, there may be as many as six flavors of rank-1 tensors or as few as one.

In the notation introduced by Sylvester in 1853, we have $\mathbf{x}=\sum x^j \mathbf{e}_j$ and $\boldsymbol{\omega}=\sum \omega_j \mathbf{e}^j$. The flavors are invariant vector $\mathbf{x}$, contravariant n-tuple of components $[x^j]$, covariant vector $\mathbf{e}_j$, invariant covector $\boldsymbol{\omega}$, covariant n-tuple of components $[\omega_j]$, and contravariant vector $\mathbf{e}^j$.

In the Penrose diagrammatic notation (similar to Cvitanovic's birdtracks and Peterson's trace diagrams), we arguably have only one flavor of rank-1 tensor, which is a vertex with one edge. (OK, I cheated. The edge is directed, so that does give us two flavors of diagrams, an innie and an outie. But if there's a metric we can reverse the arrowhead at will.)

My question: Throwing out all the historical baggage, how many of these distinctions are actually important enough that they deserve to be built into the notation? Let's assume we have a metric.