This may not completely answer your question, but I thought it might be helpful to hear a mathematician's perspective.
In modern (pure) mathematics, numbers are very important (they're the building blocks of much of the subject) but sums and products are things that were likely known to prehistoric humans and at this point they are, to say the least, extremely well understood. Thus we have moved on. We also mostly finished with calculus of a single real variable almost two hundred years ago, contrary to the curious belief among some of my previous students that calculus is the cutting edge of modern mathematics.
These days, the result of a sum is usually not a number. It could be a polynomial, an element of a Hilbert space, a vector field on a manifold, a derivation in the cotangent bundle of an algebraic variety, or a linear operator on a graded ring, to name some examples. There is also a common construction called a "free abelian group" which is us giving ourselves permission to add whatever things we want together, with integer coefficients. There would be no particular reason for it, but we could add a squirrel, a refrigerator, and a toaster if we were freeing frisky that day. The result would be
squirrel + refrigerator + toaster
unless we imposed some relations, like
2squirrel -3refrigerator + 79toaster = 0
Then the group would no longer be free, and we'd have things like
4squirrel - 6refrigerator = -158toaster
The point is that for a mathematician it often doesn't make sense to call the result of a sum a "total." For example, I'm sure we all agree that minus one hundred fifty eight toasters is not what you get when you remove six refrigerators from four squirrels. So we don't have to use a different word for this or other more serious examples, we just call it the "sum," or the "result."
That was to answer whether the distinction is made. For the rest of your question, "product" means "result" even when we aren't working with numbers. If you need an analogous word to make a distinction, I'd say call the expression the "multiplication" and the result the "product."