The mathematical names for those directions are tangent (the red arrows), normal (the green arrows, and binormal (the blue arrows). Geometers have made extensive use of these, so much so that these directions are a key part of the Fundamental Theorem of Curves. For example, see http://en.wikipedia.org/wiki/Frenet-Serret_formulas, http://mathworld.wolfram.com/FundamentalTheoremofSpaceCurves.html, and http://math.rice.edu/~hardt/401F03/ftc.pdf.

This theorem isn't of much use in orbital mechanics because torsion involves a third derivative of position with respect to time. Orbital mechanics is a study of second derivatives: F=ma.

The directions along the red arrows (v-bar) are useful in orbital mechanics because these are the directions along which you want to thrust to minimize gravity losses. The blue arrows are useful because angular velocity points in this direction. The green arrows? They're useful for geometers and for describing vehicles flying through an atmosphere. They're not so useful in orbital mechanics, which is perhaps why there isn't a standard orbital mechanics name for this direction.

The mathematical names for those directions are tangent (the red arrows), normal (the green arrows, and binormal (the blue arrows). Geometers have made extensive use of these, so much so that these directions are a key part of the Fundamental Theorem of Curves. For example, see http://en.wikipedia.org/wiki/Frenet-Serret_formulas, http://mathworld.wolfram.com/FundamentalTheoremofSpaceCurves.html, and http://math.rice.edu/~hardt/401F03/ftc.pdf.

This theorem isn't of much use in orbital mechanics because torsion involves a third derivative of position with respect to time. Orbital mechanics is a study of second derivatives: F=ma.

The directions along the red arrows (v-bar) are useful in orbital mechanics because these are the directions along which you want to thrust to minimize gravity losses. The blue arrows are useful because angular velocity points in this direction. The green arrows? They're useful for geometers and for describing vehicles flying through an atmosphere. They're not so useful in orbital mechanics, which is perhaps why there isn't a standard orbital mechanics name for this direction.

Addendum

When looking at the uncertainties in where a spacecraft is, those directions are oftentimes called along track (the red arrows), cross track (the blue arrows), and radial (the green arrows). One can look at "radial" as being either a bit of a misnomer or as being spot on correct. It's a misnomer in the sense that "radial" only points "radially" (toward / away from the planet in the case of a circular orbit. It's spot on in the sense that "radial" always points toward / away from the instantaneous center of curvature.

A related set of directions is the local vertical / local horizontal system, or LVLH for short. In this system, +Z points to the center of the Earth, +Y points opposite the orbital angular velocity, and +X completes the right hand coordinate system (i.e., $\hat x = \hat y \times \hat z$). This means that $\hat x$ points along the velocity vector in the case of a circular orbit. This labeling a bit arbitrary. The Clohessy-Wilshire equations use +X as pointing away from the Earth, +Z as pointing along the angular momentum vector, and +Y completing the right hand coordinate system. Either the LVLH frame or CW frame used to describe the orbital mechanics of a spacecraft rendezvousing with the ISS.