There are two ways to see only your downloaded songs, at least on iOS 13 and newer:

Method 1

1. Launch the Settings app.
2. Scroll down to the "Music" section and click on it.
3. Deselect the "Show all purchases" option.

This should now only show your local downloads.

Method 2

1. Launch the Music app on your iPhone and click on the "Library" icon in the bottom menu bar.
2. In the top right corner, click on "Edit".
3. A customizable list with checkmark circles should appear. One of the categories that you can select is "Downloaded", which – as the name suggests – shows only the songs that are stored locally on your phone. You'll also see another category called "Songs". This category shows all of your songs (both in the cloud and the ones stored locally on your phone).
4. Uncheck the "Songs" category and check the "Downloaded" category.
5. Click "OK" in the top right corner of the screen to close the list. You should now see only your downloaded songs whenever you click on the "Downloaded" category.

And if you want to see all your purchases once more (including those that haven't been downloaded), you can simply select the "Songs" category instead of the "Downloaded" category – that is if you haven't disabled the "Show all purchases" option in Settings.