Frequently, I have seen people use the term "order" in cryptography (the group-theoretic one). I have a mathematical background and "order" (say, for prime modulus $p$) is defined as the smallest integer, such that:
$$a^r \equiv 1 \pmod p$$
So, a generator (something which has the max possible order; i.e., order is $\phi(p)=p-1$) will have order $p-1$. In case of a composite group, the generator should have order $\phi(N)$.
How is it that, frequently, I read in crypto literature that people say generator has order $N$?
For example, the first answer at: When do we need composite order groups for bilinear maps and when prime order?
Am I missing some details or is my understanding incorrect?