'He tracked and updated 182 patient statuses' OR 'He tracked and updated 182 patient stati?'
I believe "statuses" is correct. However, the originator is claiming some sort of Latin derivation as to why this is not acceptable.
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In the scenario you've given, only a pedantic character would use Latin. Ordinary people such as medical staff and doctors would say "statuses" or refer to the patients' "medical state" and "health condition." For details, see https://en.wikipedia.org/wiki/Medical_state.
Speakers who use British Engish also use "status" as a plural noun. For example: "The survey considered participants' viewpoints, social status, and demographics." For details, see https://onlinewritingtraining.com.au/plural-of-status/.
In Latin, "statii" and "stati" are not plurals of "status." The Latin nominative and accusative plurals of "status" as a 4th declension noun are both "statūs" (pronounced 'sta-toos). This could be useful in dialog. For example:
HUFFMEISTER: By the way, saying "statuses" is wrong. The correct word is "statii."
CURLY SUE: Actually, my mother was a Latin professor. You're wrong, and you're pedantic twit.
The Latin plural of status is statūs.
A real Latin teacher would not stick Latin words into English, and so would say "statuses." If you want your character to be an obnoxious pedant but not an ignorant pedant, he or she should say statūs (typically pronounced the same as singular "status" in English).
Is your character a medical professional? He probably wouldn't say any of those things, he'd probably say, "He updated charts for 182 patients." or maybe, "He did rounds and updated charts for 182 patients." (I'm not a medical professional, but I've watched a lot of medical dramas. Also, again not as a medical professional, 182 sounds like...a lot. Even if you only see each patient for 5 minutes a day, that's over 15 hours!)