I find a definition that double indexed sum/series $\sum_{i,j=1}^\infty a_{i,j}$ converges to $A$ if for every $e>0$ there is $N$ such that $m,n>N$ $$\lvert\sum_{i,j=1}^{m,n} a_{i,j}-A\rvert<e$$
As i understand this definition implicitly assumes certain summation order - summation is done by elements in 'rectangles' and so partial sum could be defined as $$\sum_{i=1}^m \sum_{j=1}^n a_{i,j}.$$
At the same time here How to interpret a sum with two indices? is stated that notation $\sum_{i,j=1}^\infty a_{i,j}$ should be used when order of summation doesn't matter. So do I understand correctly that any usage of $\sum_{i,j=1}^\infty a_{i,j}$ and the definition of convergence mentioned earlier without specifying summation order is meaningless and not properly defined? Is this ok to say that $\sum_{i,j=1}^\infty a_{i,j}$ has several different limits? I feel totally lost in those ambiguities, can anybody shed some light on topic?
