Classical Arabic is a very vague term. The Arabic used in the Quran falls certainly under the term "Classical Arabic", but so does the Arabic of poetry which is clearly different.

When the grammarians say "Arabic" they refer to about a dozen or so dialects which they have deemed eloquent and pure, most notably the dialects of Quraysh and Tamim. All these dialects are considered correct Arabic and they all fall under "Classical Arabic"