Was the word "communist" used much in the USSR or just socialist?

The words were used appropriately.

"Communist" refered to ideology, "socialist" to economy and the current practices (and also the Socialist Bloc).

From the top of my head - Komsomol (Communist youth union) - the major Soviet youth organization. On the other hand the Union of Socialists and Non-partisans was an irrelevant spoiler organization for those few who are allowed to have place in Soviet politics without being a CPSU member.  They were used almost interchangeably one meaning another in internal affairs because there's commuist party which is socialist and who else? 

I mean in official names of organizations were they called communist or socialist?

Well, the Communist Party of Soviet Union apparently was called communist.

In Soviet Russia I remember only one party that used the official name "Party of socialist-revolutionaries" (left SRs). But it was dissolved after its attempt of anti-bolshevik revolt in July of 1918 : https://en.m.wikipedia.org/wiki/Socialist_Revolutionary_Party#:~:text=The%20SRs%20were%20agrarian%20socialists,of%20land%20to%20the%20peasants.

"Communist" was used at the names of CPSU and Komsomol. "Socialist", as far as I know, was never used, it was usually either "All-Union" or "Soviet".

The word "Communism" and the word "Socialism" have different meanings. At least in the Soviet version of Marxism. In Soviet times this difference was clearly recognised, studied in school and there was no confusion.

I would like to point out that in original Marxism, as well as in orthodox Leninism (Trotskyism), these words are often used as synonyms, because Marx did not make a significant difference, calling both communism and socialism a future classless society. This is like a whole separate debate, which I will not go into in detail now, although I am on the side of the Trotskyists here.

But in the Soviet version, socialism was called the first stage of the future communist society, when there is already a planned economy, but there are still classes or their remnants, the former norms of distribution, the state, and so on. If a capitalist state had elements of a state economy, it was called socialist elements. Those various nationalist and religious movements that also criticised capitalism and leaned towards a planned economy were also "socialist" not communist. Thus communists were specifically narrow followers of Marxism-Leninism and communism was a future classless society that was only to be built.

Thus, calling the Soviet system "communism" is an absurdity both from the point of view of Soviet terminology and even more so from the point of view of Marxism. It is a propaganda cliché, when communism was called the far from perfect Soviet system in order to identify it with the gulag, the dictatorship of the party and so on.

The word "communist" was used more often than "socialist"

Communism was a goal. Socialism was a really.

The party was called Communist because they had an objective to build communist society.

Each communist are socialist, there was no other socialists than commies in our history, so in modern Russia socialist is more likely a communist rather than a soc-dem. I mean that we don't make a distinction between commies and socialists.

Communist