The two principle components of the library are the library file, normally called iTunes Library.itl and the media folder, normally called iTunes Media. The folder containing the .itl file can also be referred to as the library folder.
The shift-start-iTunes process is used to choose or create a new library file. When you install iTunes for the first time it will attempt to open or create a library at the standard path of <User's Music>\iTunes\iTunes Library.itl, opening the file if it already exists, or creating it if it doesn't.
In your case having installed iTunes on a new system, moved over your drive, and having given it the same drive letter as before, you will need to press and hold down shift as you launch iTunes and keep holding until you see the prompt to choose or create a library. Click choose, then browse to and open the file G:\Users\Name\Music\iTunes\iTunes Library.itl
When iTunes is first installed it also assumes the media folder will be called iTunes Media, and that the media folder will be a direct child folder of the library folder. I.e. <User's Music>\iTunes\iTunes Media.
In your case you need to go into Edit > Preferences > Advanced and change the media folder to G:\Users\Name\Music. Should iTunes prompt to consolidate you shouldn't need to do this.
Your library should work in this case because the absolute file paths on the new computer will agree with those on the old one, you've taken some additional steps to ensure the drive letter is the same, and selected the desired media folder after choosing the library file.
You don't have to put in the effort to make the library portable, but if the library is in a portable shape it works on any drive letter or path, and the media folder will be set correctly after using the shift-start-iTunes method to choose the library file. A clone of the library folder on another drive is a fully functioning library that is ready to use either on the current computer, or any other running at least the same build of iTunes. Another issue, which doesn't apply in your case, is with the typical split library layout of library folder on the internal drive and media folder on an external drive or network share that isn't ready when iTunes is launched. When this happens iTunes silently resets the media folder to the default path of <Library Folder>\iTunes Media. Users often don't realise this has happened and then find some of their content isn't located where they expected, and they have housekeeping to do to set things right later.
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