I don't think we'll see much change in Cuba from 1932-1952. It was during Batista's second presidency in 1952 that the elements we associate with Cuba today really took hold.
In 1932, the current Cuban President Gerardo Machado was very unpopular and accused of authoritarian measures. Even without Batista, it is likely Machado will be overthrown in 1933 by the exact same group of conspirators (albeit without Batista, who is just a sergeant at this time). Following this, there will be a period of instability for the first year, but afterwards Cuba will be governed by democratic elections. Presumably, some other Cuban military officer fills Batista's role as leader of the military. Whether this person interferes in politics is very much based on the particular personality.
I have problems with how the link you provided seems to insinuate that Sumner Welles favored Batista and that the US supported him to quash liberal/progressive politics in Cuba. In fact, Welles did not like or trust Batista at all. In fact, the US didn't like Batista's politics and thought him a possible Communist or left wing radical early on.
US diplomatic activities in Cuba was compromised by two different goals at the time. The first was a belief that President Machado, far from being a "liberal", was in fact a dictator and the US didn't want a dictator so close to its shore. The second was a desire to not interfere and adhere to FDR's Good Neighbor Policy. Welles and the other US diplomats tried to steer their activities between these contradictory policies.
It was internal politics among the revolutionaries that managed to put Batista, a mere sergeant in 1932, into the rank of a colonel and then leader of the army which upset the normal military hierarchy at the time.
The first real change comes in 1940-1944 when Batista became President for the first time. However, he generally governed as a slight progressive and was even supported by the Communists at this time. There is not a lot of change from Cuba before his administration and after it. Regardless of who is President, we'll probably see the same policies more or less. Cuba will likely adopt the a similar Constitution of 1940 that it did IOTL.
At that point, Batista leaves the picture, preferring to live in the United States. He does not return to Cuba until 1948 when he runs as a senator.
It is only in 1952 when Batista runs for President that we really see divergences. Running a distant third int he polls, he overthrew the government in March 1952 in a bloodless coup. The previous president had become unpopular due to economic mismanagement, but he was not running for election. I have read some thing that say this was a "US backed" coup, but there is never any details. My suspicion is that this is just a lot of mythology. This seems to be mainly locally determined. Presumably, without Batista nothing like this occurs. Instead, the election of 1952 happens and some other figure, like Roberto Agramonte, becomes President. However, it is always possible that without Batista, some other military figure could rise and have a similar impact.
I would characterize pre-Castro Cuba as being relatively prosperous overall, with lots of entrenched poverty. Strong democratic tradition, but relatively weak democratic institutions. The economy was prone to a lot of boom and busts. This would likely continue without Batista. However, without Batista there would never have been a prolonged anti-government revolution or sympathy for Communists who could masquerade as being pro-democratic. Castro would never have had the opportunity to become a signficant figure or seize power. A best case scenario is that Cuba continues to develop and strengthen its institutions becoming something like a Costa Rica or Chile. Worst case scenario is a series of weak governments that become prone to military authoritarianism like much of Latin America. However, Cuba was generally much more prosperous than any Latin American country outside the Southern Cone, and once there is a strong enough middle class with some land reform, Cuba could become a very stable, prosperous country. A lot depends on particulars of Cuban politics.