The Communist Party has been active in Cuba, and supported by Moscow, since at least WWI era.
That “Communist” label was flung about by everyone.
Batista was overthrown because of his perceived (if not actual) ties to the richest of the rich upper class and those would be the ones who fled to the USA and those descendents are the ones who still scream the loudest against Castro because he took away their gravy train elitist status and wealth.