Generating sets of an Abelian group decomposed into a direct sum

Question

I’m not sure about the correctness of my assumption about the generating sets of decomposable Abelian groups, so I will be very grateful if you can tell me whether I’m right or wrong, and if I’m wrong, where are the mistakes? There is the group G: $$G = \mathbb{Z}_{12}\oplus\mathbb{Z}_{18}$$ $$\mathbb{Z}_{12}=4\mathbb{Z}_{12}+3\mathbb{Z}_{12}\cong\mathbb{Z}_{3}\oplus\mathbb{Z}_{4}$$ $$\mathbb{Z}_{18}=2\mathbb{Z}_{18}+9\mathbb{Z}_{18}\cong\mathbb{Z}_{9}\oplus\mathbb{Z}_{2}$$ The group G consists of pairs of the form $({g}_{1},{g}_{2}),$ where ${g}_{1}\in\mathbb{Z}_{12}$ and ${g}_{2}\in\mathbb{Z}_{18}$. Then the canonical decomposition: $$G\cong\mathbb{Z}_{9}\oplus\mathbb{Z}_{3}\oplus\mathbb{Z}_{4}\oplus\mathbb{Z}_{2}$$In one of the textbooks the following phrase follows: the canonical decomposition into a direct sum of cyclic primary subgroups can be written as follows: $$G=\langle(0,2)\rangle+\langle(4,0)\rangle+\langle(3,0)\rangle+\langle(0,9)\rangle$$ The group G has several different canonical decompositions, for example, the canonical decomposition will also be: $$G=\langle(4,2)\rangle+\langle(4,6)\rangle+\langle(3,9)\rangle+\langle(0,9)\rangle$$ In this regard, I have a question: is it possible to draw any conclusions about Abelian group's generating sets from its canonical decomposition? It seems to me that it is possible, because by the definition of a direct sum, each element from the group G is represented in a unique way using the sum (in this case 4) of elements, each of which belongs to one of the presented cyclic groups, and I think that as one of the generating sets, we can take one generator from each represented cyclic group (that is, from the first example of decomposition, can we assume that one of the generating sets of the group G looks like this: {(0,2), (4,0), (3,0) , (0,9)}?). And speaking about other generating sets, is it correct to say that instead of, for example, (0,2), we can take (4,2) (as taken in the second example of decomposition), because ord((0,2)) = ord((4,2)) or is there another explanation for this?