here are the common definitions of counterpoise and their references:
These come from doing a google search on this search term "define: counterpoise" copy this exactly into google without the quotes if you want to do it yourself.
<Definitions of counterpoise on the Web:
counterweight: an equivalent counterbalancing weight www.cogsci.princeton.edu/cgi-bin/webwn2.1
The second half of an antenna, which "balances" the radiating element to launch the RF wave. Although not always readily apparent, a counterpoise always exists and its suitability can greatly affect the efficiency of the antenna. A ground plane, as found on ground plane antennas, is one type of counterpoise, as is the outer case or circuit board of a receiver using an attached whip antenna.www.audiotechnica.com/glossary/
A state of equality between opposing forces.www.yourwebassistant.net/glossary/c17.htm
The reference-plane portion (grounded or ungrounded) of an unbalanced antenna.www.visionneeds.com/glossary.htm
Offset; balance; counterbalance; act as an equal effect.https://www.mises.org/easier/C.asp>
Note none of these definitions refer to a single wire. They refer to a concept. And none of the definitions refer to pattern. They do refer to efficiency. The single wire definition is simply your definition of a counterpoise. A single wire is a subset of the superset of counterpoise, but it is not the entire set of counterpoise. It is a specific example of one kind of counterpoise. As such I don't think it gets at what the question is asking. Certainly a single wire opposite the radiator in free space is a very efficient counterpoise, so is an infinite plate in free space. Note the reference to efficiency in both of these examples. You can extend the analysis to dirt, sand, rock, salt water, fresh water etc. Each of these analyze around efficiency. A silver plate in free space> salt water > radial field > good earth > sand etc. This is a very common analysis that all hams know about, and the kind of thing a body that examines hams would consider asking. So I think efficiency is the common feature being addressed in this question.
A radial system is a specific kind of counterpoise that is intended to approximate a solid plate. The shape of the radial system affects the pattern. Given these definitions I think the questions become clear. If there are great variations in the conductivity of the plate across its surface the radialted pattern takes on a shape decidedly different than circle. Since the question is specifically refering to pattern, I think the choice of the word that specifically refers to an antenna element that can affect pattern is the correct choice.
If the question added this choice "reflector" what would you choose as the answer? Remember the question expects you to pick the "best" choice. If you add the word reflector that becomes the best choice. Radial becomes the next best choice and counterpoise becomes the third best choice.
Power can effect efficiency. It occurs if the components in an antenna are under rated, where power instead of being radiated goes into melting or arcing components. Modulation can affect efficiency in the case of very narrow bandwidth antennas, but neither of these things are mentioned in the questions. Neither power or modulation affect pattern.
So this is my analysis of the questions, and I think you can clearly defend these answers before any testing body.