Yes, I think swopping over on an established site will have some potential downsides. Especially for these types of ones. 
To override the default one I think you’d need a plugin [ETA not true, see post below] (but I think the same is true for muting the notifications). They’re not editable in the UI as I think it’s easy for people to do some real damage to their site with a poorly amended one. The custom badge sql is behind a hidden site setting as an added safeguard.
I think a similar feature request has come up before too, which can add more weight to it:
(There may be others too)
A setting in your user preferences similar to the ‘Notify when liked’ one could work and keep it simple. Something like ‘Notify when badge earned’ - never | first time a badge is earned | always