Here's a bug that I forgot to report when I was doing all this research.
If you Inherit a card like Embargo, Feast, Death Cart, Mining Village, etc, and play it, it's not an Action card anymore. But Donald's current (maybe tentative) ruling is that this still counted as playing an Action card. This means that you can call Coin of the Realm, and it means that Citadel tries to replay it. I think it's fairly obvious that it's also what would seem most intuitive for most players.
It seems it works this way with Citadel (I tested with Embargo, Death Cart, Experiment and Mining Village), but not with Coin of the Realm. So Citadel replays the card, but it's an Estate so nothing happens. Nevertheless, this means that the next card I play will not be replayed by Citadel, so it makes a difference. But I don't get an option to call Coin of the Realm.
At least it should work the same with Citadel and Coin of the Realm: Either the client counts it as playing and Action card or it doesn't.