That article says:
"a function can be either reentrant, thread-safe, both, or neither."
It also says:
"Non-reentrant functions are thread-unsafe".
I can see how this may cause a muddle. They mean that standard functions documented as not required to be re-entrant are also not required to be thread-safe, which is true of the POSIX libraries iirc (and POSIX declares it to be true of the ANSI/ISO libraries too, ISO having no concept of threads and hence no concept of thread-safety). In other words, "if a function says it is non-reentrant, then it is saying it's thread-unsafe too". That's not a logical necessity, it's just a convention.
Here's some pseudo-code which is thread-safe (well, there's plenty of opportunity for callbacks to create deadlocks due to locking inversion, but let's assume the documentation contains sufficient information for users to avoid that) but not re-entrant. It is supposed to increment the global counter, and perform the callback:
take_global_lock();
int i = get_global_counter();
do_callback(i);
set_global_counter(i+1);
release_global_lock();
If the callback calls this routine again, resulting in another callback, then both levels of callback will get the same parameter (which might be OK, depending on the API), but the counter will only be incremented once (which is almost certainly not the API you want, so it would have to be banned).
That's assuming the lock is recursive, of course. If the lock is non-recursive, then of course the code is non-reentrant anyway, since taking the lock the second time won't work.
Here's some pseudo-code which is "weakly re-entrant" but not thread-safe:
int i = get_global_counter();
do_callback(i);
set_global_counter(get_global_counter()+1);
Now it's fine to call the function from the callback, but it's not safe to call the function concurrently from different threads. It's also not safe to call it from a signal handler, because re-entrancy from a signal handler could likewise break the count if the signal happened to occur at the right time. So the code is non-re-entrant by the proper definition.
Here's some code which arguably is fully re-entrant (except I think the standard distinguishes between reentrant and 'non-interruptible by signals', and I'm not sure where this falls), but still isn't thread-safe:
int i = get_global_counter();
do_callback(i);
disable_signals(); // and any other kind of interrupts on your system
set_global_counter(get_global_counter()+1);
restore_signal_state();
On a single-threaded app, this is fine, assuming that the OS supports disabling everything that needs to be disabled. It prevents re-entrancy from occurring at the critical point. Depending how signals are disabled, it may be safe to call from a signal handler, although in this particular example there's still the issue of the parameter passed to the callback being the same for separate calls. It can still go wrong multi-threaded, though.
In practice, non-thread-safe often implies non-re-entrant, since (informally) anything that can go wrong due to the thread being interrupted by the scheduler, and the function called again from another thread, can also go wrong if the thread is interrupted by a signal, and the function is called again from the signal handler. But then the "fix" to prevent signals (disabling them) is different from the "fix" to prevent concurrency (locks, usually). This is at best a rule of thumb.
Note that I've implied globals here, but exactly the same considerations would apply if the function took as a parameter a pointer to the counter and the lock. It's just that the various cases would be thread-unsafe or non-re-entrant when called with the same parameter, rather than when called at all.