I suppose it's important that, unlike many other Communist states, Cuba had it's own revolutionary past which it was unambiguously quite proud of. If you look at Cuban currency, for instance, there are only two Communist figure (Che Guevera, on the 2 Peso note, and Camilo Cienfuegos, on the 20). The rest are mostly standard Cuban nationalist figures, José Martí, Máximo Gómez, Antonio Maceo (yes, no Castro - compare this to the PRC, where each denomination is a portrait of Mao).

The "Republic of Cuba" was founded by these people, that the Cuban state and people still lionize for their struggle against Spanish imperialism, and it probably wouldn't have done much for their credibility had they essentially disestablished the state these people founded and instead replaced it with some assorted generic a grab bag of "Democratic", "People's", "Socialist", "Soviet", and/or "Worker's". In much of the rest of the world, Communists were overthrowing regimes where not only the current leaders but basically the entire constitutional order the state was founded on was seriously being questioned and could more easily be jettisoned (yet others were backed up by Soviet tanks and could do as they pleased).

It also may play a part that initially Fidel Castro (whether from honest intentions or not) more or less played coy with establishing a Communist Soviet aligned state, he initially appointed some non-Communists into the government and came to America and announced that he had no intention to establish a Communist state. The non-Communists were mostly out within a few months, America was soon giving him the cold shoulder, and it took a couple of years for him to finally nationalize foreign property. He actually waited all the way until 1965 (seven years) to officially declare his party a Communist party along Marxist-Leninist lines. So, keeping the older name could've been part of this earlier strategy that just stuck.

As for the meanings of the rest of those title's, it really depends on context. China is by far the biggest country bearing the name "People's Republic", so it's often associated with it. But, most of the satellite states of Eastern Europe also adopted it (all but Moldovia, Czechoslovakia, and East Germany). As well, a good amount of states in Africa who's wars for independence were fought by Marxist parties of various degrees of actual alignment to the Soviet union adopted it (Angola, Mozambique, Ethiopia), although they've now all dropped it. Algeria and Bangledesh use it to this day even though they were never Communist states at all (Algeria was at least single party with a fashionably revolutionary bent, but Bangledesh was always just a plain liberal democracy, it's use here is very curious).

As for "Democratic", within Marxist-Leninist terminology, they tended to use the word "Democratic" in a very peculiar way, more or less organizing masses of people towards certain goals. And, particularly after WWII, they tended to arrange large "coalitions" of parties, one ostensibly social democratic, one ostensibly populist, one ostensibly social liberal, etc... and of course, one actually Communist, into large fronts where all theoretically participated in government. In practice, of course, they were all subsumed under the power of the ruling party. And this united front list was the only thing on the ballot, you could check it off or spoil it.

There was a very large umbrella of vague, obfuscated ideas like this all labelled under the term "proletariat democracy", and contrasted negatively with western "bourgeois democracy". That it lacked free and open selection of candidates for political and government office (you know, all that any normal person would consider sufficient for accurate use of the label) seemed very much a secondary concern to them. It's notable that nearly all of the Communist states that were established in competition to some other, usually capitalist, state adopted "Democratic" in their title - East Germany, North Korea, North Vietnam, South Yemen. Perhaps they thought it made them sound more attractive in the war of propaganda.

As for "socialist", pretty much just a grab bag term for a Socialist country. The USSR was of course first in this regard. There seemed to be a pattern in eastern Europe in the 60's and 70's of switching to "Socialist", usually from "People's" (Albania, Czechoslovakia, Romania, Vietnam, and Yugoslavia did this). Bizarrely, Sri Lanka, I just found it, is fully titled "Democratic Socialist Republic of Sri Lanka" - don't for the life of me know why, again, another democratic state. And though it has now passed, the Great Socialist People's Libyan Arab Jamahiriya will always have a place in my heart for the most ridiculous, grandiosely titled country of all time.

Great Socialist People's Libyan Arab Jamahiriya 
2 March 1977-23 October 2011
RIP