Here is a scan of the October 5, 1957 announcement from Pravda:
The first sentence reads:
В течение ряда лет в Советском Союзе ведутся научно-исседовательские и опытно-конструкторские работы по созданию искусственных спутников Земли.
This can be translated:
For many years now scientific research and the construction of prototypes has been going on in the Soviet Union with the goal of creating artificial Earth satellites.
Notice that the word спутник (satellite) is not capitalized and is qualified by the adjective искусственный (artificial). We could just as well have translated the last part thus:
...with the goal of creating artificial moons of the Earth.
The New York Times in its story the same day Soviet Fires Earth Satellite Into Space called it a moon three times starting in the third paragraph:
The official Soviet news agency Tass said the artificial moon, with a diameter of twenty-two inches and a weight of 184 pounds, was circling the earth once every hour and thirty-five minutes. This means more than fifteen times a day.
So on October 5, 1957 the word "спутник" was already well-understood to the Russian reader in the metaphorical sense of a moon, an object in orbit around a larger body.
Interestingly the article in the NYT does not use the word "sputnik" once. But the next day they ran a follow-up story about the Russian word entitled Soviet ‘Sputnik’ Means A Traveler’s Traveler. Within weeks the English-language press had adopted the Russian word as a proper noun and was referring to the first artificial satellite as "Sputnik I".
This process in which a Russian word was adopted into English with only its newest meaning preserved is not uncommon. It can happen to English words too when Russians borrow them.
For example a decade or two after Sputnik I the English word "sex" entered Russian, but only with a meaning which many Americans still considered slang. Today one can use секс to refer to the sex act, but would not be understood if we were to use it to say things like "the female sex" or "the battle of the sexes".
Another example is the word сепаратор (separator) which refers to a device used in a dairy.
This answer is based in two articles which I wrote earlier: