As the sole user of a PC, which most simplifies backing up my data: when program...

I'm going to be using AOMEI's free backup program to save backups to an external drive. To set up a backup schedule, I end up drilling down multiple folders and multiple different locations and I'm always worried I'm missing something. (Why can't all my personal use data files be in one place -- rhetorical question)

My laptop recently had to have Windows reinstalled from scratch when a recent Windows update borked the whole system. (Thanks a lot MS. /sarcasm) So, I have install a bunch of programs and want to set things up to save me from headaches as much as possible.

Also curious if there are other considerations of my particular setup that suggest a preference between "for all users" or "for just me"?

The vast majority of modern apps store your data in your profile folder. The difference between "For All Users" and "Just For Me" is only in where the executables are stored: For All Users installs the program files to Program Files, whereas "Just For Me" stores the program files inside the hidden AppData folders inside your profile.

To save your data, you don't need to save the program's executable files, only your user data, and that will be in the same place regardless where you tell the application to install.

Regarding your other asides about missing data, that is either misconfigured applications, or applications that fail to store user data where it is supposed to be stored.

So, to specifically answer your question and to summarize: Either works fine. The biggest issue could be that is AOMEI is also backing up the entirety of AppData, your backups could be larger, but that would not actually be of much (or any) benefit.