I just ran into this problem myself. I had Apple Music Family, Apple TV, and iCloud 2TB. I had previously tried out Arcade (I think it was free for some period of time due to buying a new device), but didn't make use of it and didn't renew it. But it wasn't until the other day that I realized I could be paying the same amount of money for the things I did use and get some extras (that I probably still wouldn't use). Since there was a "free month" I signed up.
Several days later I got an email telling me that they charged me for my Apple Music Family subscription. Of course, that didn't make sense, so I submitted a request for a refund and provided plenty of detail of my situation. A couple days after that I was refunded. Great. But then when I went into the Apple Music app, it didn't think I was subscribed.
So, of course, I then spent far too much time talking to a rep via online chat, followed by a scheduled appointment for someone to call me the next day, which I just finished up with and involved the first rep (in the Subscriptions department *I think*) transferring me to a different rep (Technical support), who then transferred me back to a supervisor back in the Subscriptions department. I may have even forgotten about one more transfer to a different rep in the midst of all of that.
It finally got resolved by the rep telling me what Gonzo pointed out in the fine print of his screenshot in post #10 above and basically telling me to subscribe to Apple Music Family again. Because of the confusion and time wasted she was kind enough to offer me a free month of Apple Music Family (which, since even that still sounds confusing to state it that way, basically means that I'll pay $15 less for my first charged month of Apple One Premium...at least I think that's what it means).
So Gonzo is technically correct that we were warned about this (in fine print) up-front. And, I suppose, that I can even appreciate the point that maybe we shouldn't have expected to get a free month of something we were already paying for. But I'll contend that this is still a huge mess of their own making. Not only did I waste far too much of my time trying to sort this out, but it also involved time wasted by multiple people on their end who clearly didn't understand it, either. The very first interaction on my part of submitting the request for a refund coupled with all of the detail I provided (and their ability to view the specifics of my account) should have prompted whoever processed that refund (and, hence, the cancellation of my Apple Music Family portion) to send an email back to me explaining the terms of the offer and where I was clearly mistaken.