When item ids are sent to Responses API it will load them from the
database ignoring the provided values. This adds extra latency.
Not having the mode to store requests also allows us to simplify the
code.
## Breaking change
The `disable_response_storage` configuration option is removed.
This PR does the following:
- divides user msgs into 3 categories: plain, user instructions, and
environment context
- Centralizes adding user instructions and environment context to a
degree
- Improve the integration testing
Building on top of #3123
Specifically this
[comment](https://github.com/openai/codex/pull/3123#discussion_r2319885089).
We need to send the user message while ignoring the User Instructions
and Environment Context we attach.
- Introduce websearch end to complement the begin
- Moves the logic of adding the sebsearch tool to
create_tools_json_for_responses_api
- Making it the client responsibility to toggle the tool on or off
- Other misc in #2371 post commit feedback
- Show the query:
<img width="1392" height="151" alt="image"
src="https://github.com/user-attachments/assets/8457f1a6-f851-44cf-bcca-0d4fe460ce89"
/>
