I found the story here: https://www.ancientworldmagazine.com/articles/regulus-bagradas-dragon/
Searching for more information just finds the same quotes or links back to that page, which is very lacking in any conclusions as to what actually happens and just offers a few unsupported guess.
Do we have any information of what actually happened, or how people in Rome reacted when a "my troops were eaten by a dragon" story was accompanied with proof?
In Latin, serpens and draco are frequently interchangeable. That's why Thomas Aquinas in the Summa Theologica can talk about humans commanding "dragons"^1 (dracones)--he's referencing a saints' legend that is very clearly based on the staffs of Moses and the Pharaoh's magicians turning into serpentes.
So Greek δράκων, which can also mean generally "serpent," is really not a leap that signifies any sort of "dragon."
This is pointed out clearly in the Stothers article that the other piece claims to cite.
Stothers' article itself is not great, though. Just for starters, his assertion that by the early Middle Ages, "draco" and "serpens" were separate is...not entirely true, as I pointed out above.
Second, Stothers and the Ancient History piece both draw a distinction between "historical" and "literary" texts, which is...not really something modern historians would do. While there is of course a distinction between someone writing a historical chronicle and someone writing a "fiction" story, there is nothing "non-literary" about historiae. Those authors are using rhetoric and have their agendas just like 'fiction' authors.
Third, Stothers pretty much starts out with his conclusion--the accounts are true, the serpent is a python, and the size is probably exaggerated--and finds parallel mentions of too-large serpents in other ancient texts to support that conclusion. However, he's taking all of these out of their own literary and societal contexts.
And then he represents earlier scholarship disingenously when it disagrees with his point. For example, Stothers calls the work of Edward Bassett "noncommittal" on the matter. This assessment has...basically nothing to do with Bassett's actual argument.
Bassett studies specifically the account by Silius Italicus in the Punica, one of the ones quoted and discussed in the clickbait article. And he spends about 20 pages demonstrating how Silius presents this episode as a loving, wildly educated elaboration of/reference to earlier stories and authorities--the Aeneid, other Virgilian texts, the labors of Hercules.
Bassett makes a pretty good case--I mean, he gets down to the level of individual words and phrases in showing how Silius draws from earlier sources. This is a common and beloved rhetorical device in ancient and medieval literature, not something to criticize.
But while in this case, Bassett's argument concerns the elaboration/polishing of one late, "literary" account of Regulus and his serpens, we can take the underlying lesson from it. The root story reminded Silius of earlier episodes--in particular, stresses Bassett, of Hercules.
So within this tradition, there may well have been a large snake and a snakeskin in "reality." But also (or instead of?), telling this story was a way of linking Regulus and his army to heroes of yore--a way to polish his, and Rome's, prestige and majesty in historical memory.
Silius' heroicization of Regulus in his army's fight against the giant serpent finds reflection in modern scholarly discussion of Regulus as a historical/literary figure. Eleanor Leach and Rebecca Langlands, for example, are just two of the people who have highlighted Schrödinger's Regulus. In some ancient accounts, he is an exemplar of virtue; in others (especially Polybius), he is trash.
Debates about the philosophy of exemplarity, as it were, are exactly as thrilling in comparison to DRAGON SLAYING as you think.^2 For present purposes, we can say that there were both positive and negative ideas about Regulus circulating in the Roman world.
For writers who wanted to use their histories to establish Regulus in historical memory as a model, Regulus the SLAYER OF DRACONES makes him a hero through the feat itself and through the links to epic heroes--no matter how big, small, real, or imaginary the draco may have been.
~~
^1 The Middle Ages are the best ages.
^2 This is unfortunately where my familiarity with this topic actually comes from. Studying the exemplars themselves is waaay more awesome than studying the theory, and what I am there for, but some things can't be helped.
I'll respond to your original question, since it appears not to have been addressed. We don't know how the story of the encounter with a large serpent or whatever else was taken in Rome. It is a problem for the First Punic War that the most thorough surviving account is Polybius, who does not mention this event. This itself is a good reason to think that nothing may have been sent back in the wake of the Libyan campaign and that a story developed around something that was on display in the forum (some modern commentators have postulated about the sewn together corpses of elephants or something else, can't remember off the top of my head).
But, beyond the story's absence in Polybius, having only the periochae of Livy at this point means we don't get much detail. The other sources pointed out in the Ancient World Magazine article help, but don't really give us much to go on. And, sadly, there is nothing in Silius Italicus to help us gauge this. The way the story is framed in his poem is as a metaphor for the destruction of Regulus' army [found myself re-reading this and should probably correct this to be "Roman armies" more generally] (cf. the lines following the Bagradas story: "Had Regulus lived on to our time [Second Punic War period], the Trebia would never have overflowed its fatal banks with blood, nor would the waters of Lake Trasimene hide so many famous dead.")
Thus, the "unsupported guess[es]" in the Ancient World Magazine article are based on what little we know of the story, which has no conclusion (if you're looking for its effects at Rome).
Just as an aside, I think that its absence in Polybius is grounds for viewing it as a later story (perhaps circulating during the Second Punic War, at least if Silius Italicus' inclusion of it is anything to go on). Presumably the reaction would have been rather flamboyant, though not out of the ordinary. If Regulus had been triumphant and received a triumph, presumably whatever it was would have been paraded through the city. As he was defeated, though, if something had been brought back then it could have just been seen as an oddity, a strange trophy from an otherwise disastrous campaign.