The problem with this first approach is that you can get arbitrary results just be reordering the series. Also, the formula you give for the geometric series only works for |x|<1.
Talking about it as summation might be misleading since that’s one of those concrete terms that mathematicians like to redefine without anyone’s approval. Picture we have a library that includes many tricks and approaches for taking an infinite series as input and outputs a number. We know it works as expected on every convergent series. But we forgot to put in any preconditions and we’ve let people input things that are not convergent series. But whoa in many cases we are still getting a number out of it, and it’s always the same answer no matter what we do. Maybe that’s something worth studying?
It's also probably worth noting that, if the series is genuinely summable, then Cesàro summation gives its sum.