Is there a way to type something in the format "MM:SS" without it turning into "HH:MM:SS AM/PM"?

I’m trying to put in a time of 16 minutes and 42 seconds into a cell, every time I do so it turns into “4:42:00 PM” and there’s nothing I can find online to make this not happen.

Yes. At least in recent versions (I tested with 7.1), you may pre-format the cells (columns) with a format string [MM]:SS, and then those cells will accept your 16:42 as you wanted. There was ~recently some change that made cell date/time format taken into account when processing user input IIRC.

That’s the special case of a mm:ss input (versus hh:mm) which is accepted on a [MM]:SS pre-formatted cell since LibreOffice 6.4, see release notes.