Home folder name changed but terminal prompt is still showing last name
What the terminal prompt is showing you is not the home folder name, but the user name.

The folder is displayed after the machine name, in the ~ highlighted below:

name-mbp:~ old-home-folder-name$
Folder   ^ ^ User name


You can type the command pwd ("print working directory") to see which folder it is currently in. Or you can type the command echo $HOME to show you what Terminal thinks your home directory is.