Apple uses the term "Purchased" even for free apps and does mean that it was downloaded with your Apple ID previously. You should have the cloud icon with the down arrow to download the app again.
You are also able to view your Purchase History if you are curious on when it was downloaded. There are filters you can use when viewing in the App Store to restrict the last 90 days or show All. You can also redownload it from this page.
View your purchase history for the App Store and other Apple media services - Apple Support
If you are still unable to download the app, please provide a screen shot of what you are seeing.