To comment on the question here (in community-wiki mode), I should point out first that $\S13$ of my now-ancient book was meant to develop some properties of weights just in the framework of abstract root systems without having developed the representation theory where the ideas first arose.    In this experimental spirit, my exercises were a bit speculative (and in some cases not worked out by me in detail until much later, since I knew the conclusions were correct, e.g., Exercise 13.10).     

That being said, my Exercise 13.13 is intended to be done using just the tools of this chapter.  However, the exercise itself comes from one in Bourbaki, VI, $\S2$, which I've now added (reluctantly) to the informal notes linked in the question.  The Bourbaki exercise relies on their development in Chapter V of ideas about reflection groups, applied to affine Weyl groups.  This gets complicated, since it uses the dual root system in an essential way.   I didn't go into affine Weyl groups in my 1972 book but did include an account in Chapter 4 of my 1990 book on reflection groups.   An advantage of that setting is to visualize things better: the affine Weyl group leads to a euclidean simplex (fundamental alcove) whose nonzero vertices are of the form $\varpi/c$ for the fundamental weights $\varpi$ and corresponding coefficients $c$ in the highest root of the (dual) root system.   Only for $c=1$ do you get minuscule weights (none in some types of irreducible root system).     

Anyway, one is led to the list in my Exercise 13.13 (or in Bourbaki's Chapter VIII).    A subtle point is the dualization, which assigns (in Bourbaki's numbering which I adopted) $\lambda_\ell$ to type $B_\ell$ and $\lambda_1$ to type $C_\ell$.   Bourbaki originally reversed these, but that seems to be corrected in the English translation.  (Note too that they used E. Cartan's symbol $\varpi$, a version of the handwritten $\pi$ for *poids*, whereas I used $\lambda$.   Their notation is more commonly used but is easy to confuse with $\omega$.)         

Finally, a comment on the spelling of *minuscule*.   This is the original version, emphasizing MINUS (I guess in contrast to *majuscule*),
but often rendered as *miniscule* with emphasis on MINI.    The question here mixes both spellings, which is undesirable.