User logs are not sent to the console

[seeM]

Originally posted by @ChadGoymer in #4169

Is anyone having trouble displaying Python logging in the console? The code below should print messages (for warnings and above as that is the default level for the logging module). However, I do not see any output:

Contrast this with the same code, run on the same Python interpreter, in VS Code:

Is there something I am missing in Positron?

Originally posted by @ChadGoymer in #4169

[seeM]

We'll need to figure out how to configure logging so that Positron logs are separate to user logs, and so that logging behaves like the IPython or Python REPLs by default.

Since IPython already seems to handle this fine, and our kernel, shell, etc, are subclasses, we should be able to learn from how they're handling it.