There are two sides to the claim: what happened, and what's significant about it. It's not hard to come up with a variety of accurate claims that what happened in 1800 was some kind of first, but the one you pose isn't one of them. Just to address the title, "was the 1800 US presidential election the first intentional, peaceful transfer of national control?" -- of course not, for example the Emperor Diocletian abdicated due to ill-health in 305 AD. Granted there was civil war within about a year, but the actual transfer of national control from Diocletian was peaceful and intentional!
You have not correctly quoted the material you reference. What they actually claim is not, "the first time a nation's ruling faction willingly ceded authority to their opposition, simply because they agreed to do so, rather than due to threat or usage of violence". It is "the first ever peaceful transition of power after bitterly contested popular elections fought by principled partisans". So, on a point of order, if you want to ask about why historians claim the former thing, you should find an example of historians claiming the former thing and refer to that.
The claim actually made is hedged around with those extra conditions precisely in order to make it at least arguably true.
Setting aside for the moment the need to search all of history, the most obvious comparison to make is with the USA's own former colonial power, Britain. Which of course did have elections in which governmental authority was ceded to opposing factions, and so is a rich source of disproofs for incautious claims to political innovation in the USA!
British political factions of the time were certainly partisan, but were they "principled partisans", in the sense described in this material? It points out that "the Republican party had introduced a new set of principles by which the government was to be administered, very different from the Federalists' principles", and mentions Jefferson's observation that "every difference of opinion is not a difference of principle."
Were British elections "bitterly contested"? That seems to me to be a subjective judgement, and presumably some British contests were more bitter than others.
Were they even "popular elections", or is that a criterion intended to include the early US but not the Britain of the time? Recall that Britain still had large numbers of so-called "rotten boroughs". The right to vote resided not in property-owners in general, but in those owning or leasing specific eligible properties in the constituency. Old Sarum for some time in the 18th century elected two MPs, by the vote of a total electorate of 7 people: calling this a "popular vote" is a stretch even by the standards of the very limited franchise of the US election of 1800. There were constituencies ("pocket boroughs") where a single person owned the majority of eligible property, such that the electorate was whoever that person chose to install as tenants. Since the ballot was not secret, you can imagine how that might go. The land-owner could simply choose the MP.
By present-day standards we might say that none of these, 1800 included, were "popular elections", since the electorate wasn't sufficiently high a proportion of the population to satisfy us. But a difference of degree becomes a difference of type as soon as you are careful about the definitions you choose to make when you indentify your "first whatever-it-might-be".
There could also be some argument that British elections are not (or were not at the time) a full "transfer of power", because at the time the monarch still had some meaningful (albeit contested) influence on government, and the House of Lords was in the midst of government. There is one clear difference between US and British elections, which is that the US elects its head of state and upper house, whereas Britain does not. So, if that is part of your definition of "transfer of power", then Britain is neatly out of the way.
Then, having ruled out British democracy as not being of the same type as US democracy, can historians likewise dismiss other earlier modern democracies ("modern" meaning post-mediaeval)? I'm pretty sure they can, the important point is that doing so is a choice made by the person framing the claim.
I think there's a couple of things going on here. First, that 1800 US election was indeed the "first" of something, and is noteworthy in modern political history globally, not just in the US. But it takes a great deal of care to define precisely what it's the first of, other than to state the obvious and commonplace, which is that it was the first transfer of power in a republic via elections held under a US-style constitution.
Second, there may be a bit of a "rah! rah! USA was first!" attitude among some of those either learning or even teaching US history in US schools, which causes them perhaps to spend less time and effort rummaging into that detail, than someone would whose main interest was in precisly what this successful implementation of the US constitution represents in modern political history. Thus, it is easy for a carefully-framed and hedged claim to turn into a clearly false claim like, "the first time power changed hands in an election".
For what it's worth, classical democracies are generally not included in this kind of comparison at all. They have several differences from modern party politics, and also from the modern conception of a nation-state. Of course classical examples do sometimes serve as inspiration to those thinking about modern democracies, and are well worth studying, but in an effort to identify/invent milestones in modern history, it's usually not that fruitful to consider whether some Athenian official appointed by election in 450-ish BC does or doesn't satisfy our modern notion of a head of government.
Also bear in mind that in the ongoing development of ideas of democracy, there is a tendency that every progression is accompanied by claims that "at last, we have achieved proper democracy". So, Solon's democratic system in Athens is all very well, but it's not "proper" democracy until Cleisthenes reduces the built-in power of the nobility, right? And so on literally forever. Both Britain and the US have broadly extended their franchise more than once, proponents of each version believing that the previous version was not acceptably democractic in the way that the new version is. Future generations will regard our current democratic procedures as inadequate in some way or other. So in my view it's not really worth trying to make broad claims to "the first democratic such-and-such", except as a way to look in detail at what the innovations really were.
So, what actually were the innovations surrounding that 1800 election? Loosely speaking, the US constitution was the first time any nation (meaning the modern idea of the nation-state, not just any independent group of people) had created a formal position of head of state which is put up to a general vote (albeit on a small franchise by present-day standards, and with the complication of electoral college delegates), and where elections therefore can be fought on the basis of trying to convince the public that your program for government, and indeed your principles of what government should and should not do, are better than the other guy's. 1800 was the first time the incumbent lost. By no means was everyone confident the loser would leave peacefully under such a system, but he did, so the wager made by the framers of the constitution had paid off, and the notion of a modern republic was proven to be up and running. It's a matter of aesthetics how to condense that to a pithy "first-ever" milestone, and mistakes in doing so result in incorrect claims!