Rather than getting hung up on the words, consider their usage. The authors' definition may only make sense in a specific context that that they were trying to convey.
Ductile was about the ability to draw a material into a wire. The word or its roots may even preceed quantification of structural properties, but that's somewhat irrelevant. What is relevant is whether meaning gets across. For a textbook, that may mean: Use these equations when solving homework while understanding that these theoretical scenarios behave close enough to reality to be useful.
Malleable was about the ability to hammer a sheet. Physically that would mean using impact to apply large forces to cause a material to expand out of axis to the impact. While most material can expand laterally when pressed, not all will avoid shattering from the localization of the impact, and not all will retain cohesion if pressed.
Ductile was about the ability to draw into a wire. Physically that would mean tension in place of malleable's compression. Rather than area, resulting length would be the measurement.
Both of these (along with every other phrase or statement) come with unsaid caveats. For example: What is cohesion anyway? Different materials have different structural strengths and so bonds are relative. Could we then consider certain liquids to be a very malleable and ductile material so long as the molecules attract somewhat? I can start with a drop of water in the form of a bead on my anvil, and slowly press a hydrophobic hammer into it until it's scattered evenly across the entire anvil. Must be very malleable indeed. Sure I couldn't get it to retain that spread flat shape if I removed it from the anvil, but I can hammer some copper foil to the point where it would break if I peeled it off the anvil too.
The caveat that is most important that people seem to be forgetting these days: All statements are only true where they are true. If your math/model/logic are not following reality, it's not reality that's wrong. Observe reality rather than accepting statements to mean what you think they mean. Do not, for example, take that definition of ductility and some unrelated statement of rubber's elongation to mean that you can simply pull it into a wire at room temp and pressure. Verify the applicability and relation. If you can't verify before the application, leave room to react when the application of theory fails.