If you installed Budgie following the instructions in the link you have given, you should be able to select Budgie in the login screen after you have booted your computer, where you enter your user password to login. Like, select your user, and before you enter your password, there should somewhere be a menu to click on to select Budgie instead of Xfce.
Edit:
You need to click on the icon on the right side of your user name.
Edit2: on a related note. Just for testing, it might have been more convenient to install Solus Xfce to a virtual machine.
