Here they say:
First of all, since A is a discrete groupoid, the only morphisms are of the form $\text{id}:x\to x$ Also, here is the definition of discrete category in ncatlab:
A category is discrete if it is both a groupoid and a preorder.
I would say that there might be a $x$ and a $y$ with $x\neq y$ and $\text{hom}(x,y)=\star$.
So in my view there are definitely more morphisms than $\text{id}_x:x\to x$!
There's a mild disagreement on terminology here. Your first source wants a discrete category to satisfy the axiom that if there's a morphism $f:x\to y$, then $x=y$. The nLab, as usual, is a bit less demanding: their notion of a discrete category is one such that, if there's a morphism $f:x\to y$, then it is unique and an isomorphism. Thus, for instance, the category $a\leftrightarrows b$ generated by two objects with a unique isomorphism between them is discrete for the nLab.
The reason for this gap in terminology is that discrete categories in the sense of the nLab are equivalent to categories discrete in the first sense, and the nLab, along with many categorical thinkers, prefers definitions that are invariant under equivalence of categories.