Quite a bit of these can be found in the Pilot-Controller Glossary:
APRON− A defined area on an airport or heliport
intended to accommodate aircraft for purposes of
loading or unloading passengers or cargo, refueling,
parking, or maintenance. With regard to seaplanes, a
ramp is used for access to the apron from the water.
Ramp and Apron are synonymns:
TARMAC is not, by itself, defined, however there are a number of terms related to operations on tarmac, for example:
TARMAC DELAY− The holding of an aircraft on the
ground either before departure or after landing with
no opportunity for its passengers to deplane
Leading me to believe that any place can be considered "Tarmac" except movement areas.
Taxiway is also not explicitly defined (oddly, I'm sure it is well defined in the airport construction guide), but high-speed taxiway is. I assume a taxiway is the same thing except a more reasonable speed:
HIGH SPEED TAXIWAY− A long radius taxiway
designed and provided with lighting or marking to
define the path of aircraft, traveling at high speed (up
to 60 knots), from the runway center to a point on the
center of a taxiway. Also referred to as long radius
exit or turn-off taxiway. The high speed taxiway is
designed to expedite aircraft turning off the runway
after landing, thus reducing runway occupancy time.
Although I don't believe all taxiways are required to be lit. So I would say Taxiway is:
Taxiway: A defined path for an aircraft leading from runways to parking areas, hangers, gates, cargo terminals, or other operations area's.
Parking Area and Gate are not defined in the PCG, these terms should be pretty easy so I'll give a go at it:
Parking Area: Specified location(s) on the airfield where aircraft may be parked for extended periods of time.
Gate: Designated passenger loading/disembarking area for access to terminals, other gates, and airport entrance/exits.
Hardstand is also not defined by the PCG, but Wikipedia gives it a good go:
Hardstand: A hardstand is a paved area for parking heavy vehicles.
So to answer your questions directly:
Is this a binary split: Runways vs tarmac?
Yes, since you can have a "Tarmac Hold" but not typically hold on the runway, I would say that airports do not consider the runway a part of the tarmac.
Is the gate stand part of the tarmac?
Is a parking area different from a ramp?
A parking area can be different from a ramp, but is not always. For example at an FBO, the parking area and the ramp are usually the same. The parking area for airliners is probably not on the ramp though.
Is a taxiway considered part of the apron?
No, since the apron is defined and intended to accommodate aircraft for the purpose of loading/unloading, etc. These are not operations you would perform on a taxiway.
Are aprons made of Tarmac?
They can be in the sense that they can also be made of concrete, asphalt, etc.
Is a de-icing area part of the ramp?