It’s all the typesetting characters, and only if you copy and paste from another source that is most likely using a web font instead of one of the basic 5 fonts. If you type directly into the editor, you don’t have the curly quotes issue.
If I copy and past directly from a correctly displaying FR post with “ & ‘, it will convert them to incorrectly displaying characters.
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.