"The second easiest way (for 1.12.2 only) is to install the v1.2.* api-forge jar"
What do you mean by "install"? there is no installation file.
I tried adding files

baritone-api-1.2.15.jar
baritone-api-forge-1.2.15

to the "mods" folder and there is nothing happens. There is no Baritone in the game.

Final checklist

• I know how to properly use check boxes
• I have not used any OwO's or UwU's in this issue.

did you install forge
otherwise how's baritone-api-forge-1.2.15.jar supposed to work

for vanilla client.json install... 1.12.2-Baritone extract to .minecraft/versions