chore(BA-469): Rename variables api_rqst to api_request for clarity

[rapsealk]

Jira ticket: BA-469

This pull request focuses on renaming the variable api_rqst to api_request across multiple files to improve code readability and maintain consistency. The changes are primarily in functions handling web requests and responses.

Variable renaming for consistency:

• src/ai/backend/client/cli/app.py: Renamed api_rqst to api_request in the run method.
• src/ai/backend/client/cli/proxy.py: Renamed api_rqst to api_request in the web_handler and websocket_handler methods. [1] [2]
• src/ai/backend/client/cli/session/app.py: Renamed api_rqst to api_request in the run method.
• src/ai/backend/client/func/session.py: Renamed api_rqst to api_request in the stream_app_info method.
• src/ai/backend/web/proxy.py: Renamed api_rqst to api_request in multiple methods including web_handler, web_plugin_handler, and websocket_handler. [1] [2] [3] [4] [5]

Checklist: (if applicable)

• Milestone metadata specifying the target backport version

Refs

• feat(EN-115): Deprecate X-BackendAI-SSO header for pipeline service authentication #3353 (comment)