Sorry, I wasn't very clear. Mostly we use curve to mean the plot of a function, like the calculus usage that TFF gave above. (Especially when the context is extrapolation/interpolation, where we are most certainly talking about a plot of data and a best-fit curve.) So the definition might be something like: curve, all the points which satisfy some equation. (If they all happen to lie on a straight line we still refer to it as a curve, and I apologize.)
Again, just because mathematicians have used the terms like this doesn't mean they used it correctly.
They've used the word correctly in the given context, in terms of what all mathematicians understand "curve" to mean. In any language, one word may have multiple "correct" definitions - it's all good as long as everyone using the word has the same idea of which definition is being employed. You'll find the terminology used consistently in math books written in English, so this is by definition, correct usage in in this context, since everyone in the field has tacitly agreed to that usage. If they were inconsistent, it might be a mistake, but that terminology is quite consistent.
I mean, much as we may rail against changes or irritating neologisms, usage is what makes language "correct" or "incorrect". If the majority of people accept and use a word or construction, then it's correct. So, the majority of mathematicians consider any locus of points* satisfying an equation a "curve", and it is thus correct.
Edit: *Should clarify that I mean in 2-D, specifically.
