I found that the Third International in 1921 sponsored a worldwide transformation of former “socialist” parties (that was concretely related to the rejection of so-called “reformism”, to be somewhat upgraded to a more revolutionary stance, aiming to the “proletariat dictatorship”), who were thus prompted to become “communist” even in their names.
Why did not the same idea apply to the USSR, that always kept its “socialist” definition?
The reason is that there are two different concepts that are named "communism". One is the final stage in the Marxist(-Leninist?) development model (after archaic/primitive classless societies, slave-holder societies, feudal societies, capitalistic societies and socialist societies) It is marked by (again!) a classless society, a total worker's paradise etc. As far as I am aware the Soviet Union and its satellites never claimed to have reached that stage of development. They were still in the transition between Capitalism and Communism, i.e. in the Socialist phase of development.
The other concept is the political movement that works towards the establishment of "Communism, the development stage".
It is always fine to name a party after a political movement. And in fact many of the ruling parties in the Soviet bloc called themselves "communist". E.g. the CPSU.
But naming a country after some distant of what it wants to be would invite misunderstandings and derision ("Why can I not have what I want? Marx said 'everyone according to his needs'!)
It is good to know that by Lenin's stance USSR was not in fact a socialist state. Their aim was socialism and eventually communism (in modern parlance people think of socialism and communism, usually communism is just though of a subcategory of socialist views. In fact Marx himself used communism and socialism interchangeably as an economic mode), but at Lenin's time they were not even an industrial country. In fact Russia was in the verge of collapse after the WW1, civil war, economic collapse and the Allied invasion of the formed Soviet Union. Lenin thus had to be quite practical, and first order priorities were holding the country together and stabilizing the political situation to prevent further invasions by the hostile western countries and secondly to industrialize the economy in order to have resources to transform the mode of production. He openly admitted that capitalism would be preferrable to the current state of Russia. Lenin actually openly and often denied that Russia was socialist, but rather that it was their goal:
No one, I think, in studying the question of the economic system of Russia, had denied its transitional character. Nor, I think, has any Communist denied that the term Socialist Soviet Republic implies the determination of Soviet power to achieve the transition to socialism, and not that the new economic system is recognized as a socialist order
In fact USSR was first announced to have "reached socialism" under Stalin, who had - yes - greatly industrialized the Soviet Union, but not fundamentally changed the mode of production to a socialist one. Since Stalin was, before replacing Lenin a serious scholar who had analyzed much of Lenin's and Marx's writings well aware of socialist theory, it can be quite confidently be said that this announcement was made in order to advance Stalin's personal cult inside the Bolshevik party and as a propaganda message to promote Stalin's Russia. So now, officially, USSR was in fact a socialist state whose goal by the ML-definitions of the words, was communism.
I am from a former soviet country, and it is important that these ideas were somewhat new and there are some inconsistencies in the terminologies until later when they were defined more properly. Many took the word social and commune and tried to make a government definition run by social/communes. - Even Karl Marks used the words interchangeably.
So the trend from Capitalism to the paradise of Communism is supposed to happen in stages.
seize the means of production (land, labour, capital , and resources)
Establish a union state controlled by the workers.
redistribute resources each according to his need/ contribution depending on stage.
Once capitalism has been abolished and wealth has been redistributed , then dissolve into a stateless, classless, money-less, society.
The last part being important. The USSR , nor any other socialist state has ever dissolved into anything except for capitalism.
They called themselves socialist because they were in that stage of development . There is a communist party which was supposed to keep the progress moving forward to achieve the communist dream. However, Communism wasn't achieved and therefore the USSR isn't communist , but Socialist.
The Communist Party of the Soviet Union (Bolshevik) nominally advocated for an eventual classless society. Much like the reformist Fabian socialists did.
Attempts by Soviet proletarians to advance their own interests at Kronstadt or the Ural-Siberian method were not effective. The SDLP(b) or CPSU(b) foiled and fouled such efforts.
Efforts towards generalised proletarian self governance in 56 (Poland and Hungary) or 68 (Czechia of Czechoslovakia) were militarily defeated by Polish or Soviet parties.
The analysis is either: communism is bullshit. Or, that making ideological claims about what you wish is utter bullshit when you shoot proletarians to defend an ideology not built out of factory praxis. That is to say that you can put lipstick on a pig, but when you put a sausage in its mouth it will be bitten off.
The other answers have covered the meaning of Socialism in the Soviet doctrine (but see also What is 'real socialism'?). I address here the meaning of Socialism at the time of the 3rd International, and why it tried to promote a more revolutionary stance.
TL;DR: To save Marxist version of Socialism (vis-à-vis reformism/revisionism/parliamentarism/ministerialism) and to help the young Soviet Republic.
Pre-Marxist and Non-Marxist Socialism/Communism
The meaning of terms Socialism and Communism has evolved over time and in different political realities. Both notions existed before Marx came to stage, associated with Henri Saint-Simon, Charles Fourier, and Robert Owen (see History of Socialism.) Marx himself adopted Socialist/Communist view first under the influence of Moses Hess, sometimes referred to as the first German Communist, and later during his stay in Paris. Then again he was not the only one calling himself Communist - the other influential ideologues were Blanqui, Proudhon, Lassalle, Bakunin (anarchist-Communist) and others, with whom Marx often was in conflict, as evidenced by his various critical writings and events like demise of the 1st International (after a dispute with Bakunin.)
Socialism and Communism part their ways
Marx has developed his own view of Socialism/Communism, and was successful in popularizing it, which is why nowadays we mostly associate Communism with his name. During his lifetime he largely used the two terms interchangeably, and firm association of Marxism with Communism emerged mostly after the Russian October Revolution and massive propaganda by Comintern. As Karl Kautsky writes in Dictatorship of the proletariat:
They did away with the democratic institutions which had been conquered by the Russian people in. the March Revolution. Quite properly the Bolsheviks ceased to call themselves Social Democrats, and described themselves as Communists.
Kautsky represents a practical stream, which was developing in Europe - notably in France, England, and Germany - while Marx was engaged in mostly theoretical work. The Socialist parties in this countries, although often sharing Marxist views, were scoring serious achievements via parliamentary engagement - which was contrary to Marxist view that opposed participating in bourgeois politics and considered revolution (complete breaking of the existing social and economical relationships) as the only way forward. This conflict surfaced on multiple occasions - in Marx harsh Critique of the Gotha program, when the newly born German Social-Democratic Party essentially adopted Lassalean view on State socialism, in Revisionist debate that eventually led to the split of the SDP into Socialist and more extreme Communist parties, in demise of the 1st International over the debate of Marx support for the violence by the French Commune (aka Blanquism) in French Ministerialism towards the end of the century, exemplified by Jean Jaures.
Democracy vs. Terror
The 3rd International emerged after the first proletarian revolution took place in Russia (The October Revolution of 1917). The Marxist view held that the revolution in a backward country could survive only if accompanied by simultaneous revolutions in more industrialized western countries. For the reasons described above, the Western Socialists were reluctant to unleash violence, as democratic political engagement bore fruits. Moreover, as their support for the war credits in 1914 has shown, they valued progress on the national level more than abstract Marxist internationalism. Being essentially a Moscow tool, the 3rd international tried to promote the more extreme position. This is reflected in the famous Terrorism and Communism debate between "the Pope of Communism" Kautsky (favoring democracy) and Lenin and Trotsky promoting the revolutionary terror (trivia: it was during this debate that Lenin coined label renegade Kautsky, which entered Soviet folklore.)
Remark
The term Socialism in its Marxist meaning is manifested in the constitutions of the USSR and its satellites. On the other hand, western states casually referred to as "Socialist" in fact describe themselves as Social Democracies or Democratic and Social states, whereas their economic system can be described as Welfare Capitalism.
The Soviet Union kind of did brand itself as communist, as the ruling party's officials name was the Communist Party of the Union of Soviet Socialist Repubics. Whether the Soviet government was communist or not depends on whether you use the impossible, rainbows and lollipop definition of communism, or the definition that matches the reality of the ideology. The confusion might be due to the fact that the words "socialism" and "communism" tend to be thrown around and applied to a whole range of different ideologies and political-economic systems, from social democracy (which although often called a form of socialism, is really more capitalist as in this ideology, the state doesn't own the means of production and people are largely free to make profits and their own economic choices) to Marxist-Leninism, all the way to pie-in-the-sky fantasies that most socialists\communists claim as the goal of the ideology.
