The link says $n$ has a primitive root if and only if $n$ is of certain form, either $n = ap^k$ where $a \in \{1, 2\}$, $p$ is an odd prime or $n \in \{2, 4\}$
This is true.
In particular $n = pq$ or $n = prq$ where $n, q, r$ are not all equal primes are disallowed.
Other than the minor nit that $n=pq$ where $p=2$, $q$ is an odd prime does have primitive roots, the obvious question is "what do you mean is disallowed?"
A primitive root is a value $g$ where $g^k \bmod n$ can take on all values in $Z^*_n$; what the statement says is that such a $g$ will exist if $n$ is of the form listed, and no such $g$ will exist if it's not.
Yet in here discrete logarithms modulo such $n=pq$ or $n=pqr$ forms are defined and used.
Well, yes, the problem "given $g, h$, does there exist a $k$ such that $g^k \equiv h \pmod n$ (and if it does exist, what is its value)" is well-formed, even if $n$ is not of the listed form. Now, as there isn't a primitive root, there's no value $g$ for which such a $k$ will exist for every $h$ relatively prime to $n$; however that doesn't make this an invalid question.
Since generator $g$ cannot be primitive root what is the keyword for $g$ for these forms?
Well, I can't think of any better term than "generator". Any value $g$ relatively prime to $n$ will generate some subgroup; the only distinction between "primitive roots" and "everything else" is that primitive roots generate the entire group. I suppose you might insist on a value $g$ which generates as large of a subgroup as possible (which would be, for the forms you listed, of size $\text{lcm}(p-1,q-1)$ or $\text{lcm}(p-1,q-1,r-1)$), however I haven't heard of specific terminology given to elements that generate subgroups of that size.
