fix(codemode): improve prompting

[rekram1-node]

Summary

• cherry-pick the catalog signature and search changes from fix(codemode): unify catalog signatures #35452 onto v2
• adapt the internal search tool to the v2 codemode namespace split
• keep inline and searched signatures consistent, with paginated search results

Verification

• bun test in packages/codemode (255 pass)
• bun typecheck in packages/codemode
• bun typecheck in packages/core
• Prettier check for changed files