It’ll depend on what we exactly mean by, “Many stories,” but I’ll assume your sources might be referencing Francesco Giuccardini’s characterization of Lorenzo de Medici as a politician interested in preserving the Italian balance of power. Even if we engage with Giuccardini critically (he was contemporaneous and a personal confidant of Lorenzo de Medici) the characterization isn’t difficult to accept at face value: could Florence, and by extension Lorenzo and the Medici dynasty, really benefit from starting wars of conquest in Italy? Probably not, so is it all that surprising that Lorenzo de Medici’s actions, as any other Italian ruler’s actions, were a reflection of the interests and constraints which were presented to them at the time?

Stretching the issue, can we ever really attribute any period peace to the will of a single person? Can any “Great Man” change the course of history, in this or any other manner?

Without getting into the weeds of “Great Man” logic (a very popular line of thought with 19th century historians, but also very criticized from the start) an analogy I like is that the sum of all social forces is a great river, and individuals steer their boat amidst the current. I cannot recall the attribution of this metaphor (Google is unhelpful, exclusively bringing up boating advice) but I think the metaphor offers a nifty visual aid where we can place individuals like Lorenzo de Medici or Ferdinand of Naples within the wider scope of events in late 15th century Italy. It would further be interesting to read the full Britannica you offer, at it might acknowledge that Lorenzo’s overemphasized role is more a consequence of early historians wanting to find additional roles for Lorenzo analogous to his place as a great patron of the arts, rather than any factual assessment of his contribution to lasting peace on the peninsula.

Stretched to its logical conclusion, ultimately we can discern that question is not, “If not Lorenzo, then Ferdinand.” But rather, why did neither ruler want war, and why wasn’t either ruler explicitly threatened with war? Because ultimately, we must accept that if a conflict did not kick off this means that conditions to spark a conflict were just not there. In fact, the watershed moment for peace on the Italian peninsula was the Peace of Lodi, which principally involved the Duchy of Milan and the Republic of Venice, neither of which we have thus far considered in our unfortunately increasingly futile quest to find a person who was singularly responsible for peace in Italy.

By the first half of the 15th century, became clear to decision-makers on the peninsula that wars were becoming increasingly costly, destructive, and undesirable. In the previous two centuries, conflicts had often (but of course not always) consisted of a large “Bully” (like Verona, Milan, or Florence) encroaching on the freedom of a smaller or weaker neighbor, or conversely a small or weak neighbor lashing out against a “bully” on the path of encroachment (ideally supported other similarly slighted communities). While this wasn’t the only form of expansion (a natural process also took place where smaller communities “fell into” the orbit of a larger neighbor peacefully) it did mean that conquest by force of arms was often “quick and dirty,” and while communities could stage valiant defenses of their autonomy, often the larger bully came out the better after a short period of conflict. What changed in the early 15th century was that the consolidation process had advanced to the point whereby neighboring polities were equally and uneasily matched, if not in geographic size then in organization of resources and ability to project power. In the early years of the century an uneasy stalemate in the north of the peninsula between two dynastic polities (Viscontean in Milan and Scaliger in Verona), predictably erupted into a highly destructive conflict which saw the Republic of Venice enter the fray as a land-based power on the side of the Milanese to partition the Scaliger state. Around the same time, the Florentine state (at this point still a “true” republic, whatever that might mean) had seized the Republic of Pisa and while neighboring Siena would survive for another century, the Florentines were increasingly consolidated on their perch in Tuscany. In Central Italy, no political or military force emerged to wrest authority from ecclesiastical power in a government system whose bones were fundamentally unchanged since the twelfth century, while in the South the Kingdom of Naples would spiral into control of individual barons as its monarchial title was contested by foreign pretenders, effectively knocked out of the games of power on the peninsula, but also in turn insulated as no Italian player wished to spark a significant conflict involving a foreign power (a sentiment which would change by the end of the century).

Conflicts did continue to occur, but increasingly the large and precarious coalitions of independent communities built through a combination of bandwagoning and coercion which had characterized wartime coalitions in the previous century would fall before more consolidated political organizations (again, Milan, Venice, Florence, and Rome). It became clear that these larger states were no longer precarious agglomerations of city-based authorities, but instead more elaborate institutions able to project power. They were by no means complete or particularly effective political organizations, but they were much further on the road to consolidation than they had been even just a few decades prior.

Enter ecclesiastical authority. Some (like Riccardo Furbini) have argued that the increasing consolidation of secular authority prompted a reaction in ecclesiastical circles, and this reaction was to work in the opposite direction and halt the consolidation of the secular states. While the Papal State was fundamentally constructed on an oder mechanism of government where temporal authority deferred and overlapped with ecclesiastical authority, it's important to keep in mind that the other “Italian States” had been constructed by neutralizing and overlapping ecclesiastical authority, but never by entirely eliminating it. So while the Papacy’s temporal power did have practical borders (areas beyond which a Papal Legate’s authority was not greater than that of local institutions) religious authority nonetheless continued to exist in Italy; it just happened to be weaker in some places and stronger in others. In order for it to continue to exist, secular authorities needed to be at the very least stopped from expanding and consolidating.

Likewise the Milanese, who had catalyzed many of the conflicts on the peninsula in the previous century, also moved towards peace in the second half of the 1400s: a destructive dynastic conflict had substituted the long-ruling Visconti dynasty with a more precarious cadet branch by way of force of arms (this was, partially, what had brought about the Peace of Lodi in the first place). And while we must awknowledge that it was that very cadet branch (the Sforza) who would eventually look for a path to consolidating their position by way of sponsoring foreign intervention in the peninsula, in the second half of the 15th century their objective was very much still that of peace and consolidation.

The florentines too, sandwiched between the Milanese to the north (defended only by the Apennine Mountains and a few irrelevant semi-autonomous communes and principalities) and the Papacy to the South (also here only buffered by the increasingly irrelevant Republic of Siena) made the decision to push for peace, also because the Medici dynasty was making moves to consolidate power and could ill-afford distractions.

So getting back to your question, was Lorenzo de Medici important? In a way, of course he was: it was he who implemented Florence’s policy interested in preserving peace. What could Lorenzo have chosen to do differently? A slightly more aggressive stance to fortify Tuscany’s borders and move to directly absorb buffer states, perhaps. The Florentines could have tried to isolate and absorb Siena. But Lorenzo instead habitually moved to preserve external peace as to focus on internal consolidation. We must also understand that this stance was also a consequence of the socio-political conditions at the time. Even a small conflict which devolved to include any greater power (Rome, Milan, or Venice) could be very destructive, and risk backfiring. Had Lorenzo de Medici developed a greater appetite for risk, maybe the Florentine state would have pushed for some sort of war. But then Lorenzo wouldn’t have been Lorenzo, and most importantly of all, the opportunity to enter into a conflict simply did not present itself, as both religious and secular political forces moved to preserve peace.