Sometimes I seem to equate planar and planar positioning, this would be one of those times. The workspace of the robot you are analyzing is the planar position and orientation workspace - i.e. $x$ and $y$ location of the end-effector and additionally some $\theta$ angle of the end-effector. In this case, the determinant of the Jacobian exists, and I'll take the papers word for the given formula being correct. It looks like what you're saying about the fully-extended and retracted states of the arm comes from the paper, so let's dig into that.
The fully-extended state of the arm refers to the first two links being parallel to each other, where their total length is the sum of each link's length, and the retracted state is the configuration where the two links are parallel but their length is the difference of their individual link length's. So, what about this robot guarantees that these configurations will always be singular?
Intuitively, I'm not quite sure. However, it appears that in these cases, we get something like $j_{1,1} = j_{1,2} + j_{1,3}$ and likewise $j_{2,1} = j_{2,2} + j_{2,3}$, where $j_{i,k}$ represents the $i^{\text{th}}$ row and the $k^{\text{th}}$ column of the Jacobian. Thus, we lose the ability to move the arm independently in some direction as the Jacobian has linearly dependent columns.
I can write out more of the math if you would like, but its really just a couple steps of work from the paper you listed.
As an aside, when the workspace of a robot has more than 2 DoFs (especially for orienting robots), the physical intuition of singularities is harder to visualize. As CroCo mentioned, it won't do much good to look into the matter more than the Jacobian loses rank, or likewise has linearly dependent columns.