The electric field as I understand it is the very thing(the situation) that the transverse waves are propagating through and there is a direction in which these very waves move - the electric field is a part of space as your RM:AO is describing - otherwise the electromagnetic field extends throughout space which is not what you are saying.
The way I read it the electromagnetic field is imaginary - it is just the situation not the substance. The substance is energy. That would mean the concept of EMR would be a changing in the EM field(EMF). I imagine the electric field, magnetic field and electromagnetic field are dealing with situations in three different ways.
Mathematically speaking the electromagnetic tensor describes the electromagnetic field in space/time - surely the description is the situation - id est a snap shot in discrete terms. I only added this small section to add to the illustration - obviously it is less pertinent to our discussion.
The "field" is the point by point variation in the potential of the overall situation (hence; "electric potential" and "Potential-to-Affect"). It is legitimate to talk about a "PtA field" in that sense, just not a "force field". And keep in mind that a fixed magnet's "magnetic field" is not the same as Maxwell's magnetic component of EMR, although related.