I noticed myself having some trouble remembering that in 「三人」, the 「人」 is read with 「にん」, but in 「人口」, it’s read with 「じん」(and in each case wanikani seems to mark using the other reading of 「人」as wrong.)
Do the two different readings of 「人」 (「にん」 and 「じん」) have different connotations that might help me remember which one to use?
Kind of, in general though it’s just a matter of memorization and learning through experience/exposure
There are general trends that can help you remember the readings, for instance 人 as a counter for people (i.e. in 三人) takes the にん reading, and then things like nationality/race/ethnicity are often attributed with じん (ex: アメリカ人 – American)
But of course, there are plenty of exceptions, and in the end it’s best to come up with your own ways to solidify the different readings in your mind – for example adding an extra layer of a ninja being involved in the reading mnemonic story for when it’s the にん reading or something 
There’s also these entries from jisho, kind of a summation of the general trends you can see:
Also, if you search something like “人 にん じん” in the community search bar at the top right of the screen, you’ll pull up plenty of topics discussing this which may interest you ^^
Good to know, thanks 
にん are people that do something. 料理人
じん are people that are something. 暇人
that’s for 人 as suffix.
人数 and 人口 differ isofar that 人数 are generally just “people”, they’re nothing.
人口 are also people, but they’re affiliated, they are (members of a nation/town/continent).
人類 refers to people that are (part of mankind).
you could distinguish them by “born this way” (じん) and “become by choice” (にん).
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