The use of 'crayon' became muddled in the 20th century by the advent and use of Crayola crayons in 1903. Designed for use by children--safe, non-toxic--generations of children now grown understand 'crayon' to designate specifically waxy coloring sticks, that is, 'Crayola' brand crayons. In fact, however, 'crayon' (noun) was and still is used to mean
a. A pointed stick or pencil of coloured chalk or other material, for drawing.
(from the OED, italics mine)
The use as a noun, from Latin creta (chalk) via French, is first recorded in English around 1660. Likewise, the verbal use ('to draw with a crayon').
So, your idea that the use of 'crayon' meaning specifically a waxy non-toxic coloring stick [usually or always branded "Crayola"], [started out as] a regionalism, has some basis in fact: that use arose and spread with the Crayola brand of crayon. 'Crayola' itself was invented by the wife of one of the inventors of the waxy, non-toxic crayons, who compounded the French word for chalk and part of the French word for 'oily'.
For a history of Crayola crayons see, for example, "Crayola Crayon History".
Edit: Response to Comment on this Answer by OP
Your parenthetical statement in the question ("to the exclusion of colored pencils, markers, etc.") sponsored my conclusion that the usage muddle arising from the dominance of Crayola brand waxy, non-toxic crayons in the crayon market was at least partly to blame for the usage issue you encountered at work.
Your comment on this answer clarified your central concern with the usage of 'colors' and 'coloring', while at the same time acknowledging that part of the issue at work might have come from your understanding of 'crayon' as meaning specifically waxy crayons. When you clarified your concerns in your response to my answer, you defined your questions more precisely. You asked whether
- "'crayons' can be called 'colors'",
- "'coloring' is understood to be done most often with crayons (if not exclusively)".
As I point out in my response to your comment (and elaborate here), the answer to your first question ("can crayons be called colors") is yes. The OED, for example, gives the following definition under the headword colour | color, n.:
III. A coloured object, and related senses.
- a. A substance used to give something a particular colour ....
This is the sense you encountered in school. Even if the noun sense of 'color' was not as a matter of definition applied to a substance used to impart color to something, figuratively used (in synecdoche or metonymy), 'color' denotes the substance used to impart color.
Now, as I indicate in my response to your comment (and elaborate here), the answer to your second question is, with qualifications, no. Simply, 'coloring' is understood to be done with whatever substance is or might be used, whether it be wax, chalk, graphite, paint, ink or any other; in fact, no substance at all need be understood, because 'color' is a quality of light. A rainbow, for example, colors the sky.
However, I suspect your question pertains to a local, that is, a local educational, context, as given in your example ("the teacher asked her students ...."). In that case, the answer to your second question might well be yes. In a local educational context, 'coloring' may well be understood by students to mean coloring with whatever substance and instrument is most frequently or always at hand, including crayons. This situationally specific interpretation of 'coloring' is a localism (rather than a regionalism), and a narrow localism.