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You are referencing the correct article for transferring your music to a new computer. Home sharing is one of the ways to do this. It requires you to use the same Apple ID on both computers. If you continue to have issues after following all of the steps in the article you referenced, I would suggest trying the External Drive option. Unlike Home Sharing, it creates a backup of the iTunes library on an external drive.
This option requires more work than Home Sharing, but it creates a backup of your iTunes library on the external drive in addition to moving your content.