When I decline a particular update in WSUS, my understanding is that
means the WSUS server will stop offering that particular update.
TRUE. WSUS Server stops offering the declined update(s) to clients, neither for evaluation nor for installation.
However, does the Windows Update client continue to display that it
needs that particular update, but just never install it? Or do the updates
get moved to the "Optional" category in Windows Update? Or, are they simply
not displayed?
As I stated above, the Windows Update client won't be able to find the update. It'd be as if the declined update was never there. So, the client system can't evaluate the update after decline of the same.
There is also the option of UnApproving an update, and Approval for Removal of an update, instead of directly declining. You can further explore about these topics on WSUS - Managing the Updates.
Basically I'm interested in what the Windows Update client displays in
regards to declined updates, so I can communicate to sysadmins on our
network what they should expect to see if they decide to manually
check for updates.
Again, as I stated, Windows Update client won't be displaying the declined update for installation in client system. The declined updates will be only stored in the list of declined updates in the WSUS Server. Client systems won't be able to find these set of declined updates (as if they were never there).
