Nope. Allow me to illustrate:
When you click "Add" in iTunes to an album on Apple Music, you'll see these cloud symbols next to the songs, like so:
If you then click the cloud symbol, THEN that song will actually be downloaded to your drive. Up to that point, it streams (i.e., an internet connection is required) when you click the "play" button. You can right-click on the songs of the added album to also access the "Download" choice, as well as other options.
You'll notice in the top shot above that the "cloud/download" symbol appears near the album name as well. That's so you don't have to click "download" on every track -- clicking on the symbol there downloads the entire album.
PS. Carolyn Mark lives in Victoria, is a very dear friend, and something of a legend in the "high lonesome" genre of folk/country/rock. Check her out!