This article describes a concept which is mentioned in J.R.R. Tolkien's works, but was never given a definite name.
In published writing, only the term Old Noldorin was used. It was the language spoken by the Gnomes in Kortirion. In the later legendarium, the Noldor spoke Quenya both in Valinor and later in Middle-earth
Most of the corpus of ON is given in The Etymologies to describe the etymology of some Noldorin words. Even after Tolkien renamed "Noldorin" into "Sindarin", the same rules of etymology and derivation have been retained. It is therefore possible that an ancient form of Sindarin similar to what Tolkien imagined as "Old Noldorin" did exist in the First Age.
More recent works like Quendi and Eldar show ancient forms of Sindarin that seem to possess the same structure and to belong to the same evolution as the Old Noldorin of the Etymologies; but Tolkien did not give a name to that stage of language, making "Old Sindarin" to seem plausible for a name.
Note that in these recent sources, Tolkien provides these forms with an asterisk as "unattested", suggesting that they belong to an unwritten stage of the language. This is not the came in the "Old Noldorin" forms of the Etymologies where they are unasterisked.