I am totally confused by iTunes and Apple Music

OK, so I have a lot of iTunes music on my MacBook Air (Sierra 10.12.6) and on my iPhone SE2 (iOS 14.8). A week ago I started a free 3-month subscription to Apple Music. I don't have a clue how these are all supposed to work together and as usual with anything Apple I'm spending hours trying to figure it out and getting nowhere.

So on my phone I've been exploring more music on Apple Music by some of the artists I already have on my phone and computer. I click on Add to Library, and I can see the new songs on my phone. So I want to add those songs to my computer. But when I plug the phone into the computer to sync, and I click on "My device" or click on the phone icon up at the top left where you click to go to the sync screen, there is no music showing on my phone on either of those two!

So there's nothing to sync from phone to computer, and I can't drag any music from the phone to the computer! Yet the music is still actually there on my phone - I can see it and listen to it. So why isn't it there when I have the phone hooked to the computer? This has never happened before, only since I subscribed to Apple Music. Before Apple Music, when I went to sync the phone and computer, I could see all the music on both devices. Now the only music I can see is on the computer or looking directly at the phone, not on the phone screen in iTunes on the computer. I hope I'm making this clear because I am really unclear on this.

The other thing I'm unclear about is this: I infer that when I subscribed to Apple Music on my phone, that I could use Apple Music on my computer too. Is that right? But where is it on my computer? It's not in Apps. No sign of it in iTunes on my computer. And none of the songs I've added to my library on my phone are in my itunes library on my computer. But they are there on my computer when I'm signed into Apple Music, but Apple Music is a tab like any website - it's not in iTunes. So is Apple Music only on the web? Not on my computer? And not really on my phone either? Like, where is Apple Music?

But I would like to get the new Apple Music stuff on my phone onto my computer. All I know is that I can't drag the songs from Apple Music tab into iTunes on the computer. And again - when my phone is plugged into my computer to sync on iTunes, and I click on the phone icons, it shows no music on my phone. I can see the music if I look directly at the phone, but not if I look at the phone thru iTunes on my computer. Aaarrghhh!!!

I was avoiding Apple Music like the plague precisely because all Apple does is confuse me (and I've only had Apple computers since they came out and iPhones for more than 10 years). I am already really sorry I signed up. And I think that I'll just quit Apple Music altogether - if I can figure out how to do that.