The dispute about realism travels through a spectrum of views.
Most people, before engagement with philosophy of science, hold by direct realism (DR) -- that we can and do discern the reality of our world through direct perceptions. That apples are red, the sky is blue, and snow is cold. The objective world out there is directly knowable by us.
Indirect realism (IR) starts with the presumption that our perceptions (or more commonly today our experiences) are real, and we can only infer the reality of our world indirectly, and with some uncertainty. Our perceptions or experiences are still held to be confirmed through direct realism.
The work of neuroscience and the illusionist school of philosophy of mind has brought the trustworthiness of our perceptions under suspicion, and also the trustworthiness of our experiences. While the delusionists over-extrapolate off this data to call for throwing all of it out, a pragmatic alternative is to treat both perceptions and experiences as constructed inferences by our unconsciousness. This is two more options, illusionism about our interior states (IL) and pragmatism that all of our external and internal hypotheses are at least useful engineering approximations to reality (PR).
One can take this suspicion of the validity of our data a step further, and give up on the idea of realism altogether. There are variations of anti-realism, the two most widespread being pure empiricism (all we can say is that we have the data and models) and instrumentalism (the data and models are all we can really treat as valid/real) (I will label these IN).
One can take the anti-realism a step further, and deny that the external world is real at all, and arrive at idealism (ID).
The spectrum of realisms then ranges from:
DR -> IR -> PR -> IL -> IN -> ID
DR, IR, PR, and IL hold that matter is real. DR, IR, PR, IN, and ID hold that experiences or perceptions are real. DR holds that we can have certainty about matter. DR, IR and ID hold that we can have certainty about experiences.
I at least, and most philosophers of science, consider the differences between direct and indirect perceptions, and the degree of our ability to infer to an ontology, to have important considerations for our overall worldview, epistemology and ontology -- and the differences between these viewpoints to be a big deal.
As a pragmatist, I am still in the range of the spectrum considered realist. I consider that we cannot ever access "objective" reality -- and that degrees of evidence short of "intersubjective consensus supported by reams of science" are actually key to valid inferences to that reality. After all, none of our perceptions or experiences can be verified "scientifically" yet we can still make valid inferences about our selves, and our perceptions.
In the dispute you reference, Speakpigeon was advocating for the classic IR position, but with a linguistic shift where one redefined external intersubjective consensus as "objective". I objected to two aspects of this -- that we have good evidence to think that internal knowledge is less reliable than he was assuming -- and that the consequence of our NOT having access to objectivity is actually an important point that should inform our epistemology, and should not be obscured in a linguistic reshuffle. I argued the second point because I think the epistemic implications of this dispute are highly relevant to our worldviews.
Note, that most non-philosophers tend to be arguing the POVs on the left side of that spectrum, while serious philosophers tend to be arguing for the various views on the right side of that spectrum. This is why Conifold's comment implies that realism vs antirealism is where the philosophic action is, rather than in the dispute between two realists such as Speakpigeon and myself.
Now, trying to answer your questions from the POV of a pragmatist in the middle of this spectrum:
does this collective agreement imply an objective reality?
No. We have a history of collective agreements that have been overturned. So no, one cannot be certain of objective reality just because our current views have a consensus on a question.
if something is objectively the case, does it follow that people will necessarily intersubjectively agree on it being the case?
Again no. We had no agreement on the elements, before chemistry advanced a few hundred years ago. And the atomic theory of the elements is a far more recent discovery. We think these are objective reality, but for most of our history they have not been intersubjective consensus.
how can we distinguish between something being objectively the case and a bunch of individuals intersubjectively agreeing that something is the case?
We can't. We don't have access to reality, all we can do is make our best guess inferences based on our steadily improving database.
OandISAare not logically connected, so none of your statements are possibly true. Note that we can restrict the discussion to statements about natural numbers and still pick up Tarski's toolkit; it's unlikely that any sort of quibble about reality or realism is going to work, since we can talk solely of abstracta.