When it comes to vocabulary, there is only one reading that’s correct (with some exceptions). No one says 「さんじん」for 「三人」for example and you might even get weird looks if you do. So you were rightfully marked incorrect. Multiple readings (whether on’ or kun’) are usually not interchangeable. The way I would think of it is that they exist and can be the readings in other words.
When 人 is used as a suffix, there are rules (like counting = にん and nationalities = じん).
In other cases, where it’s just part of a word and not a suffix attached to a word, the determining factor is when the word was imported into the language from Chinese. にん is an older reading and じん is a newer one. But those were still hundreds of years ago, either way.
I don’t know if I’d say that you should generally remember one over the other. Rather, I would say take it as a warning sign to watch out for words that use one reading over the other and vice versa.
Yup, that is correct. In these kinds of cases, you will need to brute-force memorize. As you progress, you will generally get a feel of which reading is used thanks to patterns that you can follow, as @Leebo mentioned, unfortunately exceptions still exist.
Side note: because of said exceptions, when I come across new words, even those that contain “easy” kanji, I still need to look it up to ensure that I got the reading right.
Yeah, actually, the “simpler” the kanji concept is, the more readings it usually has, and the less confidence you can have when reading a completely new word. Kanji that are “difficult” and rarely used often have fewer possible readings.
にん and じん readings for 人 is one of the toughest problems for everyone to remember, and it hits us at the very beginning. You can throw a dice with similar success rate (slight exaggeration, of course, there are rules but they are long and full of exceptions).
Repeated exposure helps. 三人 reading was never a problem for me, since I heard it in Naruto anime hundred times as “legendary sannin”.
As for mnemonics, WK is letting us down in this case by saying “this jukugo uses on’yomi readings of the kanji” - BUT WHICH ONE???.
People rely on their own tricks, for example always using “ninja” for にん and “jeans” for じん.
I second 人’s reading being quite the hurdle in the beginning!
Just like @sansarret has already mentioned, if you’re having trouble with a specific word it may help to create a ‘second’/personal mnemonic just for that one e.g.: ‘In this 人口 everybody wears jeans.’