After some testing and playing with new code analysis settings I feel that I found out what "Show compiler errors and warnings for" and "Run background code analysis for" settings do now.
I will post my results as an answer. If you can add more details to my answer please do.
In short, the settings now divide code analysis diagnostics into two groups:
- Compiler diagnostics. Now you can configure the analysis scope for them with the "Show compiler errors and warnings for:" setting
- Diagnostics from custom Roslyn code analyzers. Now you can configure the analysis scope for them with the "Run background code analysis for" setting
The scopes are independent of each other. You can configure compiler diagnostics to be collected from the entire solution and custom diagnostic to be collected only from the current document. This way you can get build errors from the wider scope and save performance by running custom analyzers (which may require quite a lot of resources) on a narrower scope.
I tested this behavior by writing code that contains both syntax error and alert from the custom analyzer. To do this I installed xUnit unit test framework and xUnit analyzers provided with it. Then I wrote a simple unit test with syntax error and incorrect usage of xUnit:

As you can see here, the test is marked with the Fact attribute but it contains parameters. Thus, it is reported by xUnit analyzers.
Additionally, there is a syntax error in the parameters list.
My current settings for code analysis are Opened documents for both analysis scopes. When I change "Show compiler errors and warnings for:" setting to None the compiler diagnostic on syntax error disappears:

The alert is displayed again after I change the setting back to Opened documents.
And when I change "Run background code analysis for" to None the xUnit analyzer diagnostic disappears:

So it seems to me, that now you can just configure code analysis separately for serious compiler checks and custom third party diagnostics.
Still, I would appreciate if someone shared the documentation for these settings with me.