If I have multiple users on Google Chrome, and go to Settings ➜ Chrome name and picture
But for some reason, it can use my Google account's avatar's image, or a bunch of predefined images:
Is there a way to use my own picture?
You need to change the picture for the Google account to change the profile picture in Chrome. You can do that in Manage Account option.
Seems that there is no way to distinguish Google account picture from the Chrome profile picture.
Maybe you could try in Chrome profile folder, but backup data first - if Chrome will see that there are unauthorized changes, it can block your whole profile.