pwd output after moving current working directory to a new location
Assuming you're using bash, pwd is showing you the value of the PWD environment variable, which is updated when you change directory with cd. The folder1 directory changing name does not cause bash to update PWD. However you can find evidence that the directory has changed name:
pwd -Pwill show the new name of the directory.ls -l /proc/self/cwdwill link to the new name.
I think it is just the case that the first terminal has no reason to re-evaluate where it is. If you do the following command in the first terminal
cd .you will see your current working directory has indeed changed per the rename (mv).