So the constitutional framework of the Habsburg realms is a always a little messy so a subject expect might be able to correct some mistakes i make in this. But basically in 1910 the Hungarian Crownlands were a part of the broader Austro-Hungarian Empire with Franz-Joseph I being both the Emperor of Austria and the King of Hungary but since 1867 when the kingdom of Hungary a theoretically equal to Austria within the Empire and it and the broader and broader Lands of the Crown of Saint Stephen became ruled as as separate from the Austrian Empire from Budapest.

In 1844 the Kingdom of Hungary was a part of the Austrian Empire, bound into and administered from Vienna largely though the Emperor of Austria, at this time Ferdinand I was also King of Hungary as the Habsburgs had down since 1526, though historically the Hungarian nobility had attempt to fiercely guard their influence, privileges and rights with quirks of things like the inheritance of being considered a noble meaning that a pretty significant chunk of Hungarians were nobles.

As part of the waves of rebellions that occurred throughout much of continental Europe in 1848 including in other parts of the domains directly or indirectly part of the Habsburg realms and sphere of influence such as in Vienna itself, Galicia and the Italian States there rose a Hungarian independence movement that sort some degree of greater independence from Austria. The extent to which they wanted this varied between revolutionaries but ultimately supporters of the Austrian Habsburgs along with the Russians crushed these rebellions within the Habsburgs realms though it did see Ferdinand I retiring in favour of Franz Joseph I and set Hungary on the path towards the compromise of 1867.

Franz Joseph ruled the empire with a heavy hand initially in a reactionary manor to the risings of 1848 but the same pressures for a greater degree of Magyar independence remained beneath the surface. And the Hungarians weren't the only ones who were looking to assert a national identity that had shown its head in in 1848. In 1866 under Prussian leadership Germany was on its way towards unification, Austria is of course a German speaking country but the broader empire was an extremely multi-ethnic one and first language German speakers merely made up the largest minority, only about a quarter of its total. Meanwhile within the German state Austria and Germany had been competing for influence for the last few decades. Meanwhile the project of Italian unification had come on since then with the formation of the kingdom of Italy, though the Austrian Empire still held control over the the region around Venice amongst others.

This cumulated in the Prussians in 1866 in the Austro-Prussian War as the Prussians lead a coalition of Northern German states against Austria and its southern allies while the Italians joined the Prussians to attempt to take Venice while the Austrian attention was divided. In the end it proved to be a humiliating defeat that showed the cracks in the Austrian Empire's power. The Empire loosing on both fronts as it lost influence in the German states while Prussia formed the North Coalition and lost territory in Italy.

This weakness in the Austrian Empire brought back Hungary as the largest non-Austrian minority in the Empire in its push to reclaim a greater degree of independence from the crown. Franz Joseph was now in a position having been defeated where it was clear that he could no longer settle the question by force of arms alone, the Empire was too weak to achieve that. As a result he made the compromise of 1867.

In effect, the Kingdom of Hungary gained a greater degree of constitutional independence from the frame work of the Austrian Empire while Franz Joseph remained separately both the Emperor of Austria and the King of Hungary. And so there came into being the state known as Austria-Hungary in which Hungary was at leas theoretically an equal partner with Austria with a part of the Empire known as Cisleithania, which included Austria itself and a large number of other regions that had been within the Austrian Empire and Transleithania/Lands of the Crown of Saint Stephen which included the Crownlands of Hungary itself (though this was significantly larger than modern Hungary including much of modern Slovakia) and Croatia-Slavonia.

This state of affairs continues on unchanged through 1910 but then along comes the first world war. The Austro-Hungarian Empire played no small part in starting the first world war in 1914 but where men like Conrad von Hötzendorf had opened a war would allow them to reinvigorate the empire and make it and not the Russians the primary great power in the Balkans as it turns out the flames of war burned more than one eagle's wings. The empire was shattered with the union between Austria and Hungary dissolved. For a while the Hungarians attempted to hold on to as much of the modern Hungarian crownlands as possible, which included places like Transylvania and Slovakia resulting in conflicts such as the 1918-1919 Hungarian, Romanian War.

But ultimately the Treaty of Trianon saw the Entente powers back Romania who had fought on their side during world war one over the Hungarians (also in part as they did not believe Hungarian claims that it would provide equality for the ethnic minorities of the old Hungarian Crownlands) which saw Hungary stripped down to more or less its modern boarders with all of Slovakia instead becoming a part of Czech-Slovakia.

Why the language change i can't be certain. Its possible they spoke both languages of course but decided to put down the language of the nation they more identified as being from as their primary language with their identity having originally being being from the Austro-Hungarian Empire and choosing to put Austrian and German down but with the old KuK disappearing coming to more identify as a Hungarian.

I think it is worth reiterating that the U.S. census has historically been of… variable quality. I work regularly with census records from northern Indiana, and it is not uncommon for ostensibly self-reported places of birth to be inconsistent from decade to decade (along with name spellings and ages) even for native-born people. Language barriers only compounded the problems with census reporting, even though some parts of northern Indiana developed significant eastern European immigrant populations in the early twentieth century. So a poor or confused census taker is also a possibility and probably more likely than an outright lie. The instructions given to census takers in 1910 did state to differentiate between Austria and Hungary but there are no guarantees.

Out of curiosity, is there any city information? All my mom's ancestors are from Bratislava, which also gets recorded as Pressburg (its German name) or Pozsony (Hungarian). Your ancestor may have been ethnically Hungarian but lived in Slovakia, or they may be ethnically Slovak but were recorded by American census takers as Hungarian (a lot of mine were).