Distinguish webFetch and webSearch

What feature would you like to see?

<img width="560" height="393" alt="Image" src="https://github.com/user-attachments/assets/1fc48e41-f729-4309-972b-739dc7ce903e" />

This should be WebFetch('https://www.better-auth.com/docs/plugins/username'), but it displays 🌐 Searched: better-auth username plugin docs, which confuses me.

For now, there is only one Web.run tool, and the UI always show Search, which is very confused.

Why not separate it to two tools: WebSearch and WebFetch?

My current feeling is that it's not WebFetch('https://aaa.com') but rather WebSearch('site: https://aaa.com')

Additional information

codex-cli 0.63.0

config.toml:

file_opener = "cursor"
model = "gpt-5.1-codex-max"
model_reasoning_effort = "xhigh"
model_reasoning_summary = "detailed"
model_reasoning_summary_format = "experimental"
model_verbosity = "high"

[features]
unified_exec = true
streamable_shell = true
rmcp_client = true
apply_patch_freeform = true
view_image_tool = true
web_search_request = true
ghost_commit = true

[notice]
hide_gpt5_1_migration_prompt = true
"hide_gpt-5.1-codex-max_migration_prompt" = true
hide_rate_limit_model_nudge = true