Battle logs were meant to be saved and viewed so that people could put numbers to actions. It is why the battle tab is there by default.
Can you point the location to me, in the ToS, where it sanctions certain programs to view images that were taken in-game? I'm pretty sure you can't.
This whole debate is worthless and there are fallacies everywhere. The only thing you should be discussing is a lack of responses from SE. Has anyone even tried to contact a GM in-game to ask them a question about parsers? It would be much faster than waiting for a response on the forums.
On the other hand: you don't want to use a GM's response in-game. It's not very official. Just like one GM warning an account based on this where this has been the only case that has ever popped up where somewhere was warned for using FFXIVApp.
That's right: warned. Not banned. Warned.
Let's try again:
Warned. Not banned.
Do you know what most MMO companies do to people they find that are using 3rd party software? Ban them. Know what SE did in XI when people used third party software? Depends: was it Windower? Nothing. Was it a speed/teleport hack or bot? Ban. Know what SE did to people when they exploited a certain mechanic in a certain expansion without using third party software? Banned them.