chore: Migrate print-ban from test to clippy

This should be more resilient to false positives like in #12245 where a
string contains `println`.

Author: epage

Diff for: clippy.toml

@@ -1,3 +1,4 @@
+allow-print-in-tests = true
 disallowed-methods = [
     { path = "std::env::var", reason = "Use `Config::get_env` instead. See rust-lang/cargo#11588" },
     { path = "std::env::var_os", reason = "Use `Config::get_env_os` instead. See rust-lang/cargo#11588" },

Diff for: src/cargo/lib.rs

@@ -6,6 +6,8 @@
 #![allow(clippy::all)]
 #![warn(clippy::disallowed_methods)]
 #![warn(clippy::self_named_module_files)]
+#![warn(clippy::print_stdout)]
+#![warn(clippy::print_stderr)]
 #![allow(rustdoc::private_intra_doc_links)]
 
 //! # Cargo as a library