If you want more examples
(Though unclear why a mishna describing a goy shechting is not a perfect example of a goy shechting that isnt korbanos nor kosher food)
Bamidbar 14: 16 not korbanos nor kosher food
Shoftim 12:6 not korbanos nor kosher food
Melachim 1 18:40 not korbanos nor kosher food
Melachim 2 10:7 not korbanos nor kosher food
Melachim 2 10:14 not korbanos nor kosher food
Melachim 2 25:7 not korbanos nor kosher food
And none of these “shechitas” occur in the azarah so your borrowed term “distinction” doesn’t work either.
Again, I don’t doubt you can come up with creative distinctions for each (this was many people so it’s different, that was before the beis hamikdash was built so it’s different)
The simple fact is the word shechita isn’t only used for halachic shechita