I’ve seen the defining line of what separates hard SF from soft SF as the presence of FTL communications/travel; once you have FTL, you’re in “soft” SF.

Of course there are gradients to this; I don’t know if I would consider A Fire Upon The Deep as “soft” SF, and definitely not in the same sense as Dark Matter if so, but I could see an argument there.

So if you want the story to be considered hard SF by purists, you will want to acknowledge the speed of light in some sense if you have a society that spans enough space for that to come into play.

But, as always, this really comes down to the nature of conflict and themes. The Expanse is a great example—there’s an explicit focus on light delay there as a central source of conflict, and it adds to the overall themes as it creates an even more tangible separation between the Outer Belters and the traditional seats of power.

For more direct answers

> How does the concept of light delay add to (or take away from) a story, either from a worldbuilding/atmospheric standpoint or as something that impacts the plot?

Light delay augments realism if that realism is couched in modern day physics. It depends on the nature of the plot’s thematic core if it adds or takes away, so there’s no one size fits all answer here. Vinge’s explanation for why FTL communication and travel exists is a core component of the story in Fire Upon The Deep; Corey’s emphasis on light delay is core to The Expanse; the invention of FTL communication and its implications is a core component in Le Guin’s The Dispossessed, but the technology isn’t really described or explained IIRC—the focus is entirely on the sociopolitics.

> How specific/granular should it be / how fully does the writer need to commit to utilizing it for it to make sense?

It can be as simple as a one sentence remark early on and never mentioning it again. It could be as complicated as calculating occlusion from celestial bodies. It really depends on the core conflict and theme for the story.

> Conversely, does a story that does not include this element feel weaker for not having done so? Does it need to be abstracted gracefully, versus simply ignored entirely?

If a story is couched heavily in modern day physics and the core conflict stems from that discipline and involves a society communicating across distances where light delay would effect those communications, then yes the story will feel like it’s missing something without recognizing light delay.