Since you were mentioning bard performance tools when discussing "a mod is a mod is a mod", it's worth noting that so far as I know, the bard performance tools aren't a mod -- they don't modify the game process or files in any way -- they're just an external tool that generates keyboard events as if you were pressing the keys.
Functionally, it's the same as the dumb thing I tried the one and only time I tried (unsuccessfully) to perform one of my own compositions live in game; I hooked up my trusty Yamaha keyboard/synth to the computer, and wrote a simple little program that basically went "When I push this note, type 'R'." and so on. Using this, I could go into bard performance mode and play a simple little melody.
Or I could open Notepad and 'play music' into a text file, because each music-keyboard key was being treated as an alphanumeric-keyboard key. (Don't you just love the part of the melody that goes "wqe544y234te273654e"? That's my favorite!)
Now, this didn't work out well for me; my muscle memory wanted to play chords, and that gives bard performance mode indigestion. So it was a pretty failed experiment, and I tossed the little program out.
But so far as I know, the various bard performance tools do fundamentally the same thing, albeit in an automated manner -- they load a music file and then 'play notes' by generating keyboard keypresses, not by changing the game itself.
Now, there's certainly a discussion going on in the community about "third party tools" in general; a bard performance program is definitely a third-party tool that impacts how you play the game. So is AutoHotKey, a generic Windows utility for automating tasks based on a key press, which can be used to automate crafting (by virtue of having it press whatever keyboard key you've bound in-game to a crafting macro made with the game's own built-in macro system, and do so repeatedly in order to let you craft en masse while you walk away from the keys). So is Discord voice chat, as much as I hate that example being used. You could argue that fancy gaming keyboards that give you some macro keys qualify.
But none of those a mod, inasmuch as none modify the game itself.
(Now, you could argue that the Discord overlay is a mod, inasmuch as it does interpose itself into the game's process in order to render the overlay. Anyway.)
This isn't to say there might not be some which are mods you load into the game that can play the music directly. It's just that most of the bard tools I've seen floating around aren't of that sort; they're just a thing that functions as "I press the keyboard keys, music happens."
Now, whether you consider there to be a significant difference between third-party tools (e.g. things which impact your gaming experience in general but which do not themselves alter the game in any way) and mods (e.g. things that actually modify the game data files, executable code, or memory space in some fashion) is maybe an interesting debate to have. But there is a difference between them on a technical level, and I'd argue that while it's easy to say "nothing that modifies the game -- e.g., no mods -- is allowable", trying to blanket define all third-party tools as a no-go is a path to madness.
Do you have a Logitech keyboard? Presumably the Logitech G-Center or whatever the heck they call their gaming software is running, and since you can tie multiple keystrokes to a single key on that keyboard... that's a third party tool that impacts gameplay. Do you use Discord voice for callouts in your static? That's a third-party tool that impacts gameplay. (I also think it's a straw-man example that derails these threads more often than not, but...) Do you use websites that track the spawn times of NMs in Eureka zones, so you know whether or not you can get Pazuzu to pop? Those websites are -- say it with me -- third party tools, under a strict reading of the term.
Now, obviously, the devs are not going to enforce a blanket ban on third-party tools, because that would be insanity, as the definition is simply too broad to be meaningful; many people have "third-party software" that impacts their gameplay which they don't even think about -- gaming peripheral software that's automatically installed for Logitech, Corsair, or Razer devices, which lets you change the DPI your mouse movement is registered as, or whatever else -- and SQEX would more or less end up banning the entirety of the PC playerbase. Which, one assumes, is not the outcome desired by any party involved in the debate.
So it does seem worthwhile to draw a distinction between what's a third-party tool (like a bard performance thing that just presses keys) and what's an actual mod that changes how the game itself functions (e.g. stuff like the Dalamud plugin system), rather than conflating the two; even if you disapprove of both, there is still a difference between the two classes of "things that alter your game experience in some fashion".