No, we shouldn't limit our movie nights to PG-13-or-under movies.
The movie itself isn't being hosted on SE, so its content is irrelevant. In any case, ratings applied by the MPAA and FCC only count in the US, and many of our members don't live in the US, so those ratings aren't a useful metric (two different countries will use different criteria to determine different ratings that mean different things - assuming that they even bother to use ratings at all). All that matters is that we keep the commentary on the movie 13+.
We've watched R-rated movies before, and despite the 13+ guideline1, language that would never be allowed in a PG-13 film today is allowed, in moderation, on SE. Furthermore, it has been established on Meta that even users who think "obscenities" shouldn't be used on SE make exceptions for certain cases, including relevant quotations that have "obscenities" in the original material:
It is a direct quote. The poster is not using the word to make his point, he was quoting something else, and the swearing was part of the point. None of the obscenity was added.
More importantly, our site is devoted to SF&F in all its forms, and many of the franchises we discuss are rated R.
The actual content - i.e., the film itself - isn't hosted on the SE network, and the only way chat reflects the fact that we're watching the film is that we're talking about it. But we'd be talking about it anyway, even if we weren't watching it.
For the "no 14+ media on movie night, because chat is supposed to be 13+" idea to make sense, we would have to stop talking about many of the most popular on-topic franchises on the site - Aliens, The Matrix, Terminator (1 and 2), The Thing, etc.
On top of all this, the suggestion that our movie choices should be restricted to a certain ratings criteria ignores the fact that SE is a global network, and the ratings in the US are not universal. The agency that assigns ratings to movies in the US, the MPAA, has no authority outside the US. The same goes for the US' Television Parental Guidelines, which are, as the name suggests, merely guidelines, not laws. Many SF&F users are from the United Kingdom, for instance, where no television ratings system exists.
All we're doing in chat is discussing an on-topic movie that some of us happen to be watching. No 14+ material is being posted on the site or in chat2.
1 For what it is worth, the 13+ guideline appears to be merely a necessary byproduct of the fact that SE allows users to post contact info, and in the US, that makes the network subject to the Children's Online Privacy Protection Act. The idea isn't really to protect kids from naughty words, as I understand it, it is to prevent them from sharing their personal information in a way that might endanger them.
2 And even if we quote a line containing "obscenity" in chat during the event, which has happened on several occasions, it might be okay, as noted above. Even mods have used "profanity" during movie nights, and at other times as well... sometimes even a CM says a naughty word in chat.
I think the use of "obscene" words in chat, during movie nights or otherwise, may be a thing of the past (for the most part), and that's okay - but it doesn't mean we can't choose to watch movies containing such language, or other adult material (e.g., violence and gore), in our movie night events.