It seems they are using "2nd season" and "2nd cour" in different places, their X account uses "2nd cour" everywhere
https://x.com/NieRA_anime_EN/status/1771085130104229964
https://x.com/NieRA_anime_EN/status/1808561927146131865
etc, but that's kind of irrelevant.

Anyway, after noticing this also I looked a bit more and it seems they gave the next episode the number 13:
https://nierautomata-anime.com/onair/ -> in the right column 【第13話】 Episode 13
https://nierautomata-anime.com/story/ Chapter 13

The relevant guideline from the TV Bible for Anime is then this:

Sequels series (i.e. a direct continuation of the story) are added within the same entry. When the episodes numbers are continuous, all episodes are added to the same season. When the episode count is restarted (i.e. the first episode is Episode 1), the new episodes are added as a new season.
Episodes airing in a split cour format (e.g. Episodes 1-13, break, Episodes 14-26) are added in a single season. Examples: AKB0048

And not that it matters here, but TVDB also added it as part of season 1.

I didn't check the EN twitter, the info on the website seems to be recent. There wasn't a 13 on the story page either. It's fine if it's now set, I was just calling out the mods doing this on their own accord.