Seems to me like there are at least 3 different types of quotes we see here (when this converting problem isn’t happening). The slanted (or double accent quotes, if that is what they are called), the straight ones (”), and the curly ones. (I can only produce the straight ones seen above)
Makes sense what you say, though, that it well may be related to certain fonts other websites use, not being properly converted when copied and pasted here.
Agree, but as someone else pointed out, if you copy text from a FR post and paste it into the editor, you still get the crazy characters and this site is straight up Times New Roman, not Google font API. Should probably reboot the server, as someone suggested, and see if that resolves it.