To my knowledge this occurrence is due to the way animations are uploaded to the website and how they are loaded and unpacked to the character via HumanoidDescription. You will have to manually upload these animations yourself in order to have success with switching them.
Hierarchy comparison between the run and walk animation:
As far as I know, when the HumanoidDescription is running through all its descriptors and applying it to the character, the respective booleans are extracted and placed into the animate script to replace what animations should be in use. Therefore, regardless, the run and walk animation will always be in the intended order rather than the expected order.
To fix this or attempt to do so, simply reupload the animations after swapping the names of the booleans (that includes the folder - right click the folder and save to Roblox). Then, use the model ids in the fields of Run and WalkAnimation instead and try again.
There are no errors, and I know for a fact that it runs because I am spawned, but only the humanoid description is changed and not the animate script. I still move as the old walking animation.