Wheniwasyourage
Ooh-er. Sorry Missus, but I'll call them what I want! And while I might use the term "quotation marks", I certainly wouldn't call them "quotes".
Sorry if you are offended but you are of course, free to call them what you like. 🙂. This is, after all, “Pedants’ Corner”. 😉
The terms “quotation marks” and “quotes” have been in use for many decades since keyboards came into our lives.
A person dictating a letter would also say “insert quotes” and “remove quotes” around specific parts.
I doubt if anybody under about 40 would know them as “inverted commas”.