I did delete the starting capital for 2 reasons:

1. It was in a bad position
2. The version notes from Frogboy said that this was now possible

When I put the capital in a new location, I chose the capital icon that I usually see for core worlds.  I was not aware that there was a different kind of capital available.  In fact, as I look at the list of improvements in my current game, I do not see a Civ Capital icon.

I am going to start a new game and see if I can figure this out.

What I would recommend is that the game REQUIRE that you have identified one of your core worlds as the Civilization Capital for your civilization by the end of your turn or a message pops-up to remind you to select one of your core worlds as your Civilization Capital.  That world's capital would be the Civilization Capital improvement and the capital for all other core worlds would be the Core World Capital improvement.

This would allow players to move their Civilization Capital by just choosing the core world that they want to be their new Civilization Capital at any time during the game.  If a player destroys their Civilization Capital improvement (like we did) or if our Civilization Capital is captured, we would be prompted at the end of our turn to select a core world as our Civilization Capital or, if a player only has one core world as they do at the beginning of a game, the game would automatically make that world the Civilization Capital.

The notion that the game would allow you to proceed without identifying a capital for your civilization is not a good one.