Digital Signature Algorithm signature creation?
It depends how you define what a "public key" is. 

Typically it is the value of the key itself ($y$), plus information about the group (safe prime $p$, subgroup size $q$, generator of subgroup $g$). For signing a message, you do not need the value of the public key itself. So if you are strict in defining a "public key" to only be $y$, it is not needed to sign (only to verify). 

On the other hand, you do need the group description ($p,q,g$) to sign a message, as well as the secret key ($x$). If you are liberal and define "public key" to include the group description, then the public key (or parts of it) is needed to sign. 

I am not sure what is meant by PU$_\mathrm{G}$ in the diagram, but I suspect it is the group description? In that case, Stallings is being liberal in his definition of a public key in that it includes the group information ($p,q,g$).

As an aside, even though you do not need the public key, you do need $g$ and $x$ and since $y=g^x$, you do "know" the public key at signing time even if you do not use the value $y$ explicitly.