Is there any distinction between these two verbs or can I use them interchangeably?
Ich drückte die Klinke herunter/hinunter.
As I understand they both mean
to press down.
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The words are synonyms. But as a native speaker I mainly use "herunterdrücken" ("Ich drückte die Klinke herunter."). It is just more common / often used, and sounds more natural to say. Also "hinunterdrücken" is a word, that sounds a bit old, like it was very common hundreds of years ago.
I would only use the word "hinunterdrücken" in combination with "dort" if that "down-place" had some special meaning in the previous context and I want to point out the connection to that.
Example of that special case: "Don't do it! At this point the handle breaks. – I'll push the handle down there!" translates to "Tu es nicht! An diesem Punkt bricht die Klinke. – Ich werde die Klinke dort hinunterdrücken!".
Using "herunter" in this context would be wrong. It would translate to "I'll push the handle from down there." which doesn't make any sense.
You can see the frequency (~"Häufigkeit") of a word used in modern German in the "Duden" (the Oxford English Dictionary for German). If you can't decide which word to use, I would always go with the word which has the most "Häufigkeit" (or flip a coin if the bars are the same).