The distinction comes from modern academia in the West and isn't found historically in the languages of the region, where terms like "Syriac" traditionally referred to the same thing as Aramaic. This is reflected not only by Arabic سرياني but also by Aramaic languages themselves, where Western Neo-Aramaic speakers continue to call their language Siryōn and Central & Northeastern Aramaic languages exhibit native names like Surayt, Suret, etc. To my knowledge only Jewish communities preserved a name derived from Aram for their language, but only marginally, and the convention hasn't survived: Jewish NENA languages nowadays are referred to by terms relating to Judaism or by expressions like "our tongue".
Modern academic interest has changed things a bit by introducing terms like Arōmay to WNA and آرامي to MSA. In this light you could maybe restrict سرياني to Classical Syriac, but it's very plausible that the reader wouldn't recognize the intent if it isn't explained clearly.
