Patch 1.04 doesn't want to install on the demo.
By the way, the demo runs on a single-core processor with 256 MB of video memory, but with "artifacts" (I played like that for a week). Buying a new (budget) video card completely solved the problem.
Go for it, comrades!

Added 6 minutes ago
And what is the difference between "Skirmish" and "Network Game > One Computer (Local)"?

Skirmish is a battle between a player and the computer.
Hotseat (one computer) - you can play on one computer for several players (you can also add the computer). In Heroes 6, the settings are a bit confusing, so if you want to play against yourself, choose this "one computer" mode, delete the AI players, click "+" - add - and select the first option (add a human player). To play as a Faction, you need to create a hero for it in the dynasty menu.

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Wow! It connected me! And it didn't even reset the bonuses!
Oh, no. The bonuses were reset, but the level remained.

Ment, you were very lucky. It also connected for me, but the level is one, so everything was reset.

The game started lagging! Also, now, before the Ubisoft logo when the game starts and the desktop when the game closes, the game shows a black screen and thinks for 15 seconds. No new maps were found; it seems that this time they fixed the balance and added bugs. The version is new (1.4.****)