How to delete songs from your iPhone?

Like so that they didn't show up in your music app with the download symbol next to it so it was just gone

You'll have to delete them from any device connected to the cloud. So like if your mac has those songs on it, you would have to delete it off of the mac. Or you could just disable cloud.

You can uncheck the songs on your iTunes library first, and then sync your iPhone with iTunes, the songs will be deleted from your iPhone. If iTunes not worked for you, you can use some third party software like this iOS file manager. It can help you delete songs, videos and photos from iPhone.