server/webui: cleanup dual representation approach, simplify to openai-compat (#21090)

* server/webui: cleanup dual representation approach, simplify to openai-compat

* feat: Fix regression for Agentic Loop UI

* chore: update webui build output

* refactor: Post-review code improvements

* chore: update webui build output

* refactor: Cleanup

* chore: update webui build output

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Co-authored-by: Aleksander Grygier <[email protected]>