But stars (and gas clouds) orbiting in a disk are essentially always "near the edge" of the disk (that is, the disk defined as matter interior to the orbit in question), so the approximation is usually pretty good.

reference to the "radial velocity" or speed of stars where I believe they are actually talking about the orbital or tangential velocity.

The (understandable!) confusion originates because "radial" and "tangential" refer, in an observational sense, to velocities along or perpendicular to the line-of-sight vector. To simplify a bit: for observations of astronomical objects, velocities along the line of sight from you to the object (detectable from Doppler shifts in its light) are "radial", while velocities perpendicular to the line of sight (which require that you wait long enough to see things change position on the sky) are "tangential". For edge-on spiral galaxies, the orbital motions that we can observe are those producing Doppler shifts along the line of sight (which are, in the frame of the galaxy, "orbital or tangential", as you suggest).

And lastly, if the dark matter obeys the same laws of gravitation, there must be a reason for its distribution to be different than the other matter.

The usual argument is that dark matter is "non-dissipative" -- that is, it can't lose orbital energy via things like radiation, the way ordinary matter in the form of gas can. (And, unlike gas, it doesn't feel pressure forces.) Stars are also non-dissipative, but stars form out of gas, so their distribution reflects the effects of dissipation in ways that dark matter can't.