Quick and simple:

 

Is the hubble constant the rate at which galaxies are moving away from us or the rate of "cosmic expansion" and the speed at which the distance of space increases between us?

"Hubble's law describes the observation in physical cosmology that the velocity at which various galaxies are receding from the Earth is proportional to their distance from us." from wiki http://en.wikipedia.org/wiki/Hubble's_law

 

Hubble's law is observation.

"cosmic expansion" is an explanation of Hubble's law.

The Hubble constant is the rate of cosmic expansion.

That is funny.

Please refrain from further attempts to hijack this into a discussion of the validity of the big bang theory and cosmic expansion. Any discussion along those lines belongs in speculations

Hum.

From wiki, the whole paragraph:

Hubble's law describes the observation in physical cosmology that the velocity at which various galaxies are receding from the Earth is proportional to their distance from us.[1] The law was first formulated by Edwin Hubble in 1929[2] after nearly a decade of observations. The recession velocity of the objects was inferred from their redshifts, many measured much earlier by Vesto Slipher (1917) and related to velocity by him.[3] It is considered the first observational basis for the expanding space paradigm and today serves as one of the pieces of evidence most often cited in support of the Big Bang model.

 

Did I say anything different?