Define best_iteration only if early stopping is used. (#9403)

* Define `best_iteration` only if early stopping is used.

This is the behavior specified by the document but not honored in the actual code.

- Don't set the attributes if there's no early stopping.
- Clean up the code for callbacks, and replace assertions with proper exceptions.
- Assign the attributes when early stopping `save_best` is used.
- Turn the attributes into Python properties.

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Co-authored-by: Philip Hyunsu Cho <chohyu01@cs.washington.edu>

tests/ci_build/lint_python.py

@@ -37,6 +37,7 @@ class LintersPaths:
         "demo/rmm_plugin",
         "demo/json-model/json_parser.py",
         "demo/guide-python/cat_in_the_dat.py",
+        "demo/guide-python/callbacks.py",
         "demo/guide-python/categorical.py",
         "demo/guide-python/feature_weights.py",
         "demo/guide-python/sklearn_parallel.py",