Fix spawning code-server within code-server (#4590)

* Remove extra VS Code CLI spawn

We already spawn VS Code's CLI when necessary in the lines below.
Having the CLI spawn unconditionally when in a VS Code environment makes
it impossible to run code-server within code-server (for example to
develop it).

* Update VS Code

This sanitizes our environment variables so code-server does not always
think it is a child spawn.

Fixes https://github.com/cdr/code-server/issues/4519.