According to this Wikipedia article the USSR was socialist. But no private property was allowed and everything was controlled and owned by the government both of these were communist ideas.
Classical Marxism says that socialism is a middle ground between capitalism and communism.
And as far as I know, the USSR was the embodiment of far communism.
So how should I interpret this Wikipedia page, I'm a little confused now.
One of Engels's and Marx's famous predictions was that the proletarian revolution would not create communism, but it would create the conditions for communism. The worker-run state would triumph globally in this scenario and the state would "wither away" as it was now superfluous in this new socio-economic epoch. So "communism" was the end result of both the revolution and securing it.
All of the Eastern bloc countries during the Cold War were run by Communist parties, but both in state rhetoric and internal conceptualizations, they used the term socialism to describe their socioeconomic system. As good Marxists, they felt that "communism" was something that would arrive after a period of transition. Where the USSR had an impact here was that the Bolsheviks pushed a line that socialism could only be achieved with a Marxist-Leninist political party acting as a midwife to historical development. The nomenclature for these vanguardist parties soon became some variation of "communist" because it differentiated them from other socialist parties (some of whom were quite at odds with Lenin and company) and signaled that the much talked about revolution was now underway.
The continued existence of sundry details like money and wages were regrettable compromises that needed to be made for this utopian future, as did the existence of the state and a party to control it. Communism's arrival in the not too distant future was something that Soviet and Eastern bloc state propaganda emphasized, but state propaganda was always clear to differentiate between existing socialism and future communism. The state was all-powerful under socialism, but it simply did not need to exist in the future when communism had arrived. Khrushchev, for example, predicted that communism would arrive by 1980 in the Soviet sphere, a concrete position that soon became the butt of jokes and something his successors backed away from. Even Khrushchev hedged his rhetoric in this declaration, claiming 1980s communism would be accomplished "in the main," but the final communist utopia of Marx-Engels would still have to be built by that future generation and its successor.
So it is much more useful to think of the term "communist" not in the socioeconomic sense of a society where there is no want or money, but rather as a political terminology. In this case, referring to communism in Poland or Germany, for instance, is to discuss the political situation of their communist parties, not how much it cost a Syrena or Trabi.
The nature of the Soviet society has been a hot-button issue within the Marxist movement for decades. The three most notable leaders of the three most notable Marxist tendencies during the 1950's all had their own takes on the issue. Stalin, of course, defended the idea that the Soviet Union was, in fact, a socialist society:
The Union of Soviet Socialist Republics is a socialist state of workers and peasants.
Leon Trotsky, a former leading member of the Soviet state, held that the USSR was indeed a socialist society, but was in retreat from socialism and towards capitalism, thanks to the "parasitic bureaucracy." He wrote plenty on this, but I believe this to be a more or less acceptable introduction to Trotsky's thoughts on the USSR.
Amadeo Bordiga, representing broadly the ultra-left position, considered the Soviet society to be merely an augmented form of capitalism. In 1952, Bordiga wrote a response to Stalin's Economic Problems of Socialism in the USSR. Stalin's work was meant to defend the Soviet Union from accusations of "state-capitalism", Bordiga's writing was probably the most notable response from the state-capitalist camp. It was only recently translated into English, however, and so maybe some of Raya Dunayevskaya's writings would be a better introduction to this camp's thoughts.
It's unfortunately impossible to give a politically neutral response to this question, and so I'll merely respond how I know how to: with my own thoughts. To me, the Soviet Union was a state-capitalist society. Allow me to explain why.
As you point out, private-property did not exist in the Soviet Union (that's not entirely true actually, but that's far too complicated to deal with here). However, the existence of private-property is not the defining characteristic of capitalism, nor is its absence the essence of communism. Marx famously stated in the Communist Manifesto that "the theory of the Communists may be summed up in the single sentence: Abolition of private property." Something that is often missed, however, is that the Manifesto wasn't a great philosophical work, but was a pamphlet to action. That isn't to say that it's a bad work, far from it, there's a reason that Marx and Engels decided to republish it in the 1870's, but that it bears markings of its intended purpose. The crudeness of this line is one of those markings. The most direct refutation of this line probably comes from Engels:
I am familiar with capitalist production as a social form, or an economic phase; capitalist private production being a phenomenon which in one form or another is encountered in that phase. What is capitalist private production? Production by separate entrepreneurs, which is increasingly becoming an exception. Capitalist production by joint-stock companies is no longer private production but production on behalf of many associated people. And when we pass on from joint-stock companies to trusts, which dominate and monopolize whole branches of industry, this puts an end not only to private production but also to planlessness.
Here, Engels makes explicit that private-property isn't to be identified with capitalism. While less explicit, Marx would correct himself numerous times:
The capitalist epoch is therefore characterized by this, that labor-power takes in the eyes of the laborer himself the form of a commodity which is his property; his labor consequently becomes wage-labor. On the other hand, it is only from this moment that the produce of labor universally becomes a commodity.
Instead of private-property, Marx and Engels came to identify wage-labor, the commodified form of labor, as the root distinction of capitalism. Communism, then, was the absence, or transcendence, of the institution of wage-labor:
Within the co-operative society based on common ownership of the means of production, the producers do not exchange their products; just as little does the labor employed on the products appear here as the value of these products, as a material quality possessed by them, since now, in contrast to capitalist society, individual labor no longer exists in an indirect fashion but directly as a component part of total labor.
With this understanding, Marx, instead of targeting private-property, began to target commodity-production (since for labor to be bought and sold there must first also exist other commodities that can be bought and sold, or else for what purpose would anyone enter into wage-labor relations?) and the property that enabled it in his analysis:
Two characteristic features distinguish the capitalist mode of production from the start. First, it produces its products as commodities .... Commodity constitutes the dominant and determining character of its product. This first of all implies that the laborer himself appears only as the seller of commodity and thus as free wage laborer, and therefore labor appears as wage labor in general .... Secondly, what specially distinguishes the capitalist mode of production is the production of surplus value as the direct objective and determining motive of production. Capital produces capital and this it does only in so far as it produces surplus value
And, so, the question here isn't to determine if private-property existed in Soviet society, but to determine if commodity production and capital existed in Soviet society.
Firstly, as to if commodity production existed in the Soviet Union, the answer is unambiguously: yes. Stalin himself admitted as much in his 1952 work Economic Problems of Socialism in the USSR, and other Soviet officials had fessed up to this in the journal Under the Banner of Marxism as early as 1943. There certainly is no debate over this fact.
In our country, the sphere of operation of the law of value extends, first of all, to commodity circulation, to the ex-change of commodities through purchase and sale, the ex-change, chiefly, of articles of personal consumption.
By commodity production, it is meant the production of readily reproducible, useful objects intended for sale on the market.
Secondly, as to if capital existed in the Soviet Union, this too is relatively clear-cut. Again, Stalin and the Soviets admitted to as much, justifying this under the propagandist title of "socialist accumulation." Regardless of if it was accumulation in the name of socialism or not, it was still capital accumulation; intentions don't change reality.
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