I'll try to comment your statements and provide some clarification:
They oppose coaster brakes with free wheels.
In the article they compare two alternative braking mechanisms.
My understanding is that the opposite of freewheel is fixed gear
A freewheel and a fixed gear are indeed two different mechanisms, but it's hard to precisely define them as being opposite. In the article, because a fixed gear is not a very efficient braking mechanism, they don't consider it as a good alternative to hand brake.
the definition of freewheel is that you can roll the rear wheel while the pedals are at rest (not rotating).
This may not be the complete definition of a freewheel.
You can do that with a freewheel, indeed. But also with a coaster brake. Some people use "coasting" and "freewheeling" interchangeably when referring to this functionality. But the mechanism in a coaster brake that allows your pedals to stay at rest is different than the one in a freewheel.
This video explains how a coaster brake works:
That is, most normal bikes with coaster brakes also have a freewheel setup.
Coaster brakes can "freewheel" in the sense of allowing the pedals to stay at rest, but they don't require a ratcheting freewheel mechanism like the one in a bicycle freewheel.
what's the word for a pedaling backwards setup?
In a freewheel there's no additional mechanism for allowing you to pedal backwards, it's just that the pawls only engage in one direction, the same mechanism that allows coasting/frewheeling. So a ratcheting mechanism may be considered a pedaling backwards setup.
Edit: Or you may be confused whether or not pedalling backwards is also a functionality of the general freewheel term. Yes it is. It fits into the definition of a driven shaft spinning faster than a drive shaft. For example pedalling backwards at 60 RPM while the bike is standing still: driven shaft (rear wheel) 0 RPM > drive shaft (pedals) -60 RPM.