[Disclaimer: I am not a native speaker myself, and my english sucks as well. I posted my reply not because I enjoy being picky, but because it served my purpose of inviting people to try and think more abstractly about mathematical objects.]
Yes. Without the s and assuming one variable as the function, it’s just a partial derivative.
But even adding the s, such description could be misleading: generally, reading variable people do automatically think about a dependent variable (and not the function). I bet that a lot of people construed it as rate of change of an independent var wrt one of the others.
Another point worth to stress is, like I was mentioning, that the gradient is not just the list of all partials, regardless the fact we can hendle it as such for our present purposes.
It is a vector, e.g. an element of a vector space, enjoying a lot of unique properties. It is useful, imho, not to think about vectors just as arrays (lists of scalars), since it spoils the mindset of those who desire to dig deeper in math.
You know, If I was a teacher, I would never give a linear algebra course using the coordinate isomorphism, at least not in the beginning. It hinders people in understanding the gist of vector spaces, and the mappings between them.