Recall that
How did we derive these axioms shown above from the usual strong axioms? How do we know what words or properties to omit when creating the weak axioms? Who invented the weak axioms? How "weak" are the axioms and why are these axioms "weak" rather than normal strength? What do we mean by weakness?
I am not sure your source for those "weak" axioms, but it is clear that they posit the existence of sets that contain certain elements but more elements may exist than the minimum required. That is, for example, a pair set exists, but may contain more than the two required elements. This is a problem if you require a pair set to contain exactly the amount of elements required. The weaker axioms imply the strong ones, though, which is an exercise. See the MSE question about this topic.
The "weakness" appears to be that the "stronger" versions of the axioms clearly imply the weaker ones, but in fact, turn out to be equivalent. Some people prefer to use minimal axioms to construct a theory reasoning like Strunk and White "omit needless words". The exact choice of axioms for a theory is an art because there is no canonical choice, although there are many popular choices. Euclid's choice for geometry was popular for thousands of years.
For reference you can look at Thomas Jech's Introduction to Set Theory, 3rd edition, page 12, exercise 3.7 which gives the weak version of the three axioms.
