[Chore] Rename Processor to InputProcessor

[DarkLight1337]

Purpose

Avoid confusion with IOProcessor and OutputProcessor, and also with HF's own AutoProcessor classes. Later on, Renderer will probably contain both InputProcessor and IOProcessor.

This PR keeps back-compatibility for the old import in case people with custom engines use it.

Test Plan

Test Result

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[gemini-code-assist]

Code Review

This pull request is a refactoring that renames the Processor class to InputProcessor to improve clarity and avoid naming conflicts. The changes are applied consistently across the codebase, including in tests, engine components, and entrypoints. A new file vllm/v1/engine/input_processor.py is introduced, and the old vllm/v1/engine/processor.py is converted into a backward-compatibility shim that issues a DeprecationWarning. This is a good practice for managing such refactorings. The changes appear correct and thorough. I have not found any issues of high or critical severity.

[chatgpt-codex-connector]

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[DarkLight1337]

Flaky test