The part about being a democracy until 1944 is one of those "technically correct" claims as long as you ignore the role that Admiral Horthy played in Hungary from 1920 to 1944. Horthy's role essentially continued the former powers of the 'King of Hungary' (aka the Habsburg rulers) with powers to appoint or dismiss Ministers, and dissolve the legislature at will. I'd characterise it probably as a continuation of the former Habsburg state which was authoritarian with some trappings of democracy (an elected parliament etc), but calling it a democracy is a bit of a stretch.

The lack of choice in joining the Axis is a bit more complicated - it's certainly a rather favourable interpretation of Hungarian foreign policy, but Horthy was always a somewhat reluctant ally of Hitler, and given the Third Reich's reaction to Yugoslav rejection of the Axis, it's plausible at least to see the regime as finding itself with hard choices between the Germans and likely failed resistance. Given the history of Hungary in 1918 and 1919, and the general anti-communism of Horthy and his regime, they certainly weren't going to willingly join the Soviet camp.

The impression I get from what you've described is that it is "lying by omission" in that the claims it makes aren't necessarily untrue, it's just they leave out all the things that would contradict them

As NestorNotable said, it's technically true that Hungary was a democracy until 1944 as there were elections and a parliament. However this system was heavily skewed in favour of the governeor Horthy's party. To do a quick review of the state of democracy, before 1918 about 6% of the population had voting rights, basically the wealthy educated males. In 1919 after the democratic revolution free and secret elections were scheduled for 1920, but this never took place. A decree in 1922 by PM István Bethlen revised this, with the reasoning that the masses are easy to manipulate and such an election can only lead to chaos. Still compared to the dual monarchy it gave much broader rights, allowing males with 4 finished school classes and over the age of 24 to vote, and women with 6 classes to vote, meaning 2,5 million could vote in the country of 9 million. However this was seen too much, so with the exception of capital and larger towns, ballots were open, which as you can imagine lead to a lot of coercion, blackmailing and intimidation to vote the "right way".

This cemented Horthy's party's dominance, consistently winning over 60% in elections. In 1938 secret ballots were universally introduced, but as you expect now eligibility criteria was further tightened, so despite the change in ballots the governing party rose to 70% (!). It should be noted here that at the time Horthy's party wasn't seen far-right but as a national conservative, and they made great pains to deline themselves from the upcoming far right parties, prominently the Arrow Cross party, whom they harassed even more than leftist parties (as they were seen a greater threat due to the general emergence of far-right in 30s Europe). And ironically secret ballots helped the far-right surge in elections.

With regards to Hungary having no choice, the prime motivation behind Hungarian foreign policy was to revise the Treaty of Trianon, which led to appeaing whichever great power seemed most willing. Most Hungarian hopes were with Mussolini's Italy who was seeking new allies in the region, though there were also decent relations kept with Great Britain. A large change came with the authotorianesqe Gömbös goverments election in 1932. Gömbös was among the first to congratulate Hitler on his election to chancellor, personally visited him and Mussolni. Though Gömbös died in 1936, this foreign policy path of warming up to Italy and Germany continued. You have to see here that Hungary did indeed have little choice if it wanted to gain back territories: The revision of Trianon was seen an absolute must, the Trianon broders as unacceptable and temporary. Britain was far away, relations with France were bad. Relations with USSR was improving but the starkly opposite ideology made the soviets an uncanny partner. And there was Italy and Germany in the neighborhood, with their own shares of woes and feeling of historical injustice. When Germany started to carve up Czechoslovakia, Hungary stepped up in hopes of regaining lands, which did happen in the first Vienna awards in 1938 (I want to stress this relationship was not an alliance, which came later). For Hitler Hungary was little more than an extra pressure on the Czechs and not much of a big help.

This process started a give and take reaction. Hungary slowly got some former lands back, in exchange they had to give Germany more leeway and help. For example after 1st Vienna Hungary had to allow the nazis to organize among Hungarian Germans, had to leave the League of Nations, had to give some economic concessions etc. But in 1939 graf István Teleki became the PM which led to some changes and resistance. First when Germany attacked Poland Hungary refused to participate or even help, and allowed Polish refugees to come. Then they began to repair relations with Britain and reassured them of Hungary's neutrality. At any cost Teleki wanted to stay out of the war since he recognized Germany will lose it eventually. Now this was quite a contrarian opinion at time; At the time there was a general awe of the German political and war machine, and much of the military's leadership urged to join the war in hopes of territorial spoils and a better place in the coming nazi world order.

Then in 1940 the USSR annexed Bessarabia and Hungary started to look at getting Transylvania back. Long story short this almost escalated into a Romanian-Hungarian war, when Hitler and Ciano stepped in and negotiated new borders much to the shock of Romania. The cost? More compulsory exports to Germany, legalising the Arrow Cross party, releasing far right politicans, total control over Hungarian German political life, new law against Jews, and joining the Tripartite pact. As the final straw, Teleki signed a treaty of eternal friendship with Yugoslavia in late 1940, only to join in the Axis attack in 1941, leading to his suicide and Hungary joining the war. The following years were characterized by increasing economic dependence on Germany, though the 1942-43 Kállay-governement did upkeep secret diplomatic channels with the allies in hopes of preserving regained territories after the war, but as we know today the Germans basically knew about every Hungarian move, not the least because of the many fanatic supporters of nazis up to the highest echelons. Still, a Hungarian politican noted after the war, "out of every 10 German demands, we rejected 9". Hitler likely wanted no less than total integration of Hungarian economy and resources into the Nazi war machine, and until the German occupation of Hungary in 1944, a degree of soveregnity and independence was upheld.

So after reviewing all this, did Hungary have a choice? This is a hotly contested topic among Hungarian historians so theres no way to make a right conclusions. The policy of revisionism was an all-dominating topic in interwar discourse and politics. Was it aggresive? Perhaps, but its hard for us to put ourselves back there. I read lot of articles newspapers and publications from the era, to say that there was a seething sense of injustice and feeling that a huge wrong has been done to the Hungarian nation is no hyperbole. From this PoV, Hungary's shift towards Germany (the only country able to willing to give back lands) is not suprising. In retrospect of course it's esay to judge it knowing what Germany did under the war, but recall that Hungary warmed up to and later joined the Axis during Hitler's ascendancy, at a time when few thought Germany could lose the war. NestorNotable says well that there were lot of hard choices, manifested if in nothing else but the suicide of one of the prime ministers. But over time small concessions gradually and surely pushed Hungary into Germany's camp, as a price for the regained lands.

The alternative was to not follow up on 15-20 years of revisionism when chance came, to stay neutral (and likely occupied) in an Axis-dominated Europe (at least until 1944), to resist the huge German fanboyism prevalent in the military and social elite, to somehow keep up independence despite strong economic ties. It would have been a very harsh route, though some Hungarian leftist thinkers after the war did comment that if Hungary did so we might have actually got some lands back for our resistance, but that's obviously just a what-if.