For example, I cannot visualize a 13-sided polygon accurately, but I can draw one pretty easily without having any drawing skills.
A more realistic problem in the case of the bicycle drawing has to do with the procedural knowledge they use for drawing. For instance, maybe they approach it analytically, top-down, breaking it into major components to draw first, and those into sub components. Well, in order to do that, they need explicit knowledge of that hierarchy; they may have an accurate representation of the image somewhere, but they can't bring it directly into consciousness, and they didn't break it into that hierarchy while they had it around. So now what's left to them is to attempt to produce the hierarchy from an imperfectly recalled image, which leads to the strange configurations.
Just a guess—but I wouldn't be surprised to learn it's something on those lines.
However, if you are using an fMRI, as in the paper, all you need is the accurate representation somewhere (and you need to know where that somewhere is of course).
Another example: let's say you look at a picture of cheetah. Now you try to imagine it; what appears to you is a vague sort of resemblance but nowhere near as clear as when you were looking directly at it. But, just because the prompting for you to imagine the image didn't cause an accurate recreation in consciousness, doesn't mean it didn't occur elsewhere, perhaps at some earlier/lower-level stage of visual processing.