https://www.jstor.org/stable/2273786?seq=1#page_scan_tab_contents Is the article where it is from. It seems to have never been added to the library, which would be my fault.
https://projecteuclid.org/download/pdf_1/euclid.pl/1235417266 is the first chapter of the textbook where I personally found it, although I would use the JSTOR article in the library of Cantor's Attic after this post.
The massive textbook I used is called "Model-Theoretic Logics" by the "Ω-Group" (the coolest pen-name for a group of mathematicians). In Part F, this equivalence is proven. However, I won't get into the proof, but rather the definition of "logic" which is used.
Chapter XVII is "Set-theoretic definability of logics" (written by Väänänen, one of my heroes) which is where this comes from. The definition is quite nuanced, but it is a great read. I recommend this textbook.