On the main site I would like to have the articles with the author and the publishing date. But now way to set the publishing date to visible. Only the creation date will be displayed. I try to change the settings on the Home menu. Have you any idea?

With the Beez3 - Default Template it works!

Please check > Global settings of the Content Manager:
Show Create Date = YES
Show modify Date = YES

If I do that, the articles shows me the Create Date. But I prefer to have the publishing date.
Note, your template shows only the Creation Date. Also the modify Date is not working.

The fist Picture

is with the Beez3 Tamplate and you can see all 3 date fields (create, modified and published)

And on the 2nd one

with your X2 template you can see only the created date.

It depends on the Warp framework, it has only one date function.