I've always considered it a mistake to refer to "%" as a "modulo" operator in C because of its ability to return negative numbers. That's not how modulo arithmetic works. C's "%" is a remainder operator.
We should be a bit careful with what one means by "modulo arithmetic" here. If we are talking about arithmetic in Z/nZ (read "integers modulo n"), then the objects being acted upon are no longer integers, but equivalence classes of integers. That is, the set of all integers that satisfy the equivalence relation "~" where "a ~ b" means a - b = k*n for some integer k. For example, one of the equivalence classes in Z/3Z would be the set {..., -4, -1, 2, 5, ...}.
Since every element of the set is equivalent under that relation, we typically will choose a representative of that equivalence class, e.g., [2] for the previous class. If we view the "mod" or "modulo" operator to be a mapping from the integers to a particular representative of its equivalence class, there's no reason that negative values should be excluded. [-1] refers to the same equivalence class as [2], [32], and [-7]. The details of how integers are mapped to a chosen representative seem to vary from language to language, but modular arithmetic works all the same between them.
But 2 mod 3 should still give the same as -1 mod 3. If the representative for the same equivalence class is allowed to be different depending on the input, you could just as well use the number itself.
At the risk of being snarky, I should certainly hope the output depends on the input.
On a more serious note, there are quite reasonable properties that are satisfied by keeping the sign of the first argument as the chosen representative. In particular, for any two nonzero integers a and b, we have that:
a = (a / b) * b + (a mod b)
Where division should be interpreted as integer division that rounds towards zero. If a=-1 and b=3, then (a/b) would be zero which would require (a mod b) to be -1 if we want the above to hold. Also note that other rounding choices (e.g., rounding towards negative infinity) could impose (a mod b) = 2.
So choosing a particular representative comes down to choosing what properties you want your function to have in relation to other arithmetic.
Neither of them is wrong. Languages (like python) which only return a positive number are returning the least positive residue of division under the modulus. That is the most common interpretation of it in number theory at least far as I know.
C generally does return negative numbers when you use the % operator.
Under modulo math, -1, 9 and 19, for example, are all the same element of the modulo 10 congruence. 9 is called the "least positive residue", which is sometimes important.
"operator" is being used as a term of art in programming language design, to describe a symbol used to indicate that an operation should be performed on one or more operands.
So the "%" symbol is an operator in C and Python, and it is standard usage to describe the operator that performs a mathematical function "X" as the "X operator".
As sibling comments have mentioned, "operator" here is being used in the programming language sense of the word, not the mathematical sense. Using the name "mod" [as in mod(a,n)] or "modulo" for the mapping that takes integers to a particular representative of its equivalence class in Z/nZ ("integers modulo n") doesn't seem like an unreasonable choice to me.
(mod N) is an operator which indicates that the arithmetic in the preceding formula or equation is understood as taking place in a modulo N congruence.
A triple equal sign is usually used for modular equations. That triple equal sign, together with the (mod N) out to the right, constitute an operator.
-1 ≡ 9 (mod 10)
Since it is written as part of the syntax of a formula or equation, and modifies its semantics, it is an operator. Just like d/dx, sigma notation and whatever not.
We should be a bit careful with what one means by "modulo arithmetic" here. If we are talking about arithmetic in Z/nZ (read "integers modulo n"), then the objects being acted upon are no longer integers, but equivalence classes of integers. That is, the set of all integers that satisfy the equivalence relation "~" where "a ~ b" means a - b = k*n for some integer k. For example, one of the equivalence classes in Z/3Z would be the set {..., -4, -1, 2, 5, ...}.
Since every element of the set is equivalent under that relation, we typically will choose a representative of that equivalence class, e.g., [2] for the previous class. If we view the "mod" or "modulo" operator to be a mapping from the integers to a particular representative of its equivalence class, there's no reason that negative values should be excluded. [-1] refers to the same equivalence class as [2], [32], and [-7]. The details of how integers are mapped to a chosen representative seem to vary from language to language, but modular arithmetic works all the same between them.