Invertibility as Criteria for a Loop

Question

I try to understand the correct criteria for a Loop.
I see in Wikipedia
https://en.wikipedia.org/wiki/Inverse_element#In_a_unital_magma
that “A unital magma in which all elements are invertible is called a loop”
Also in the answer to this question:
What's the difference between a monoid and a group?
Loop “is equivalent to asking that conditions 2 and 3 be satisfied, but not condition 1” meaning Existence of neutral element and Existence of inverses are sufficient for a Loop.
However here:
https://ncatlab.org/nlab/show/quasigroup
“Note that, in the absence of associativity, it is not enough (even for a loop) to say that every element has an inverse element (on either side); instead, you must say that division is always possible” Can someone tell if there is any inaccuracy in one of the sites above or that I miss something?