The default display override is taking over and doing that. Probably doesn't make sense to have that for that field. Will remove for a new build release sometime today. Only need default display override for canvas, avatar, and formatname fields and will remove the others.

But if in the configuration is set name format to "Name only" the last name is still shown there.

That controls what the Formatname field displays. That field is used externally outside of profile so it will have a default display override of showing the name, but it won't link to profile if privacy is blocking profile access. You can use the layout parameters to better customize what Formatname field displays as well.

PS: I would wish to have a further function for the last name like in Linkedin: to set the last name public only to the first letter.

We've no functionality for this. You'd need to implement that yourself probably using a CB Code Field to use PHP to only return the first letter of the last name. It might be a simple addition though since we've a global function for formatting name. Have added a feature ticket to look into this for a future release.

forge.joomlapolis.com/issues/8203