Falling as defined by classical physics is simple
Any movement on an object due to gravity as defined by newtons laws.Not complicated, you can add direction if it helps understanding like "fall in", "fall down", "fall sideways" and you can even "fall up". The fall down option is the one most of us immediately think of but that like all things human is simply due to familiarity. Some layman and some historic scientists argued that you can't "fall up" so a very long time ago a specific term was created which is "free fall" (
http://en.wikipedia.org/wiki/Free_fall). When layman say falling scientists translate it to "free falling" which is the generic case and safe to do.
An object in the technical sense of free fall may not necessarily be falling down in the usual sense of the term. An object moving upwards would not normally be considered to be falling, but if it is subject to the force of gravity only, it is said to be in free fall. The moon is thus in free fall.
So there are many things in the universe "free falling" which we would never describe back to a layman as "falling" because they lack the basic understanding, their learning is dictated by what they have experienced.
GR turns that on it's head in that there is "no free falling" you are simply being accelerated because of the distortion of spacetime geodesic. In the clip he substituted that to "falling" because of the target audience and it does not change the meaning or result of the experiment shown.
You are correct you couldn't pick the difference with both theories they are viable options given your data so far.
Both theories predict movement and with movement you encounter wind resistance which will effect the feather more than the bowling ball, that whole mass to surface area thing. So the whole discussion about the air becomes irrelevant and specifically what we are questioning is "Uniform gravitational field without air resistance" hence the need for the vacuum.