I was under the impression that the way it's defined in the title was the correct way to interpret the UPP(as that was what I was searching for) from the definition here: https://mathoverflow.net/questions/235062/unique-product-groups-and-semigroups
but in my searching I found that diffuse groups are considered to have the UPP, and that any free abelian group is diffuse: https://www.math.uni-bielefeld.de/documenta/vol-21/24.pdf
on the wiki for free abelian groups:https://en.wikipedia.org/wiki/Free_abelian_group#Examples_and_constructions you can see that there's clearly more than one way to reach a given dot in the image on the right, and it says that "The integers, under the addition operation, form a free abelian group with the basis {1}", which seems to imply that it's some weird form of linear dependence, and therefore not the same as what I was looking for.
So is my understanding of the UPP correct, or am I barking up the wrong tree? Some of it is out of my depth, but only because I was looking to study more about a group,ring,etc. where all elements of a set can only be constructed by two other elements and the operator(perhaps making an exception for identity).
It looks like you may have skipped over the quantifier $\exists$ at least one element that can be uniquely written as a product etc. etc.
It didn't say all elements could only be written as products in one way.
So, $\Bbb Z$ would have the property, because given any two finite subsets, there's an element that can only be written as a product (in this case sum) in one way.
Let's play around with it for a moment. Take $A=\{0,1,2\}$ and $B=\{0,1\}$. Then $3$ can only be written as a product (sum) of elements of $A$ and $B$ in one way ($2+1=3$).