I'm fairly sure that: 1. Beren was indeed the one who first named her "Tinuviel" and 2. "Luthien" is a Sindarin name.
I can't recall sources for either of those two assertions at the moment, but I'm fairly sure that they exist. I'll check HoMe for them later.
I'd say that "he called her by her Elvish name" is just a bit of misleading poetic license. It's a true statement, as far as it goes; it just omits to mention that, in fact, both of her names are Elvish and that the name he called her was his own invention.
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