fix(llm): use correct binary name for VS Code Insiders detection

[yrobla]

Summary

DetectedLLMGatewayClients() uses exec.LookPath(cfg.LLMBinaryName) to verify a tool is actually installed before reporting it as detected. The vscode-insider entry had LLMBinaryName: "code" — the stable VS Code binary name — instead of "code-insiders", which is the actual CLI shim shipped by the Insiders build.

On a machine with only VS Code Insiders installed (no stable VS Code), LookPath("code") returns not-found, so thv llm setup silently skips the Insiders integration even though the tool is present.

• Fixes LLMBinaryName for vscode-insider from "code" to "code-insiders"
• Adds TestRealClientConfigs_LLMBinaryNames as a regression guard that asserts the binary name for every LLM-gateway-capable tool in supportedClientIntegrations

Fixes #5089

Type of change

• Bug fix
• New feature
• Refactoring (no behavior change)
• Dependency update
• Documentation
• Other (describe):

Test plan

• Unit tests (task test) — TestRealClientConfigs_LLMBinaryNames passes and would have caught the bug
• E2E tests (task test-e2e)
• Linting (task lint-fix)
• Manual testing (describe below)

API Compatibility

• This PR does not break the v1beta1 API, OR the api-break-allowed label is applied and the migration guidance is described above.

Does this introduce a user-facing change?

Yes. thv llm setup now correctly detects VS Code Insiders on machines where only the Insiders build is installed.

Special notes for reviewers

The regression test checks the binary name for all five tools that have LLMBinaryName set. If a future entry gets the wrong name, the test will catch it immediately rather than requiring manual verification on a machine with that specific tool installed.

[Copilot]

Pull request overview

Fixes VS Code Insiders LLM gateway detection by correcting the CLI shim name used for exec.LookPath, and adds a regression test to guard LLMBinaryName values for real client configs.

Changes:

• Update VS Code Insiders LLMBinaryName from code to code-insiders so detection works when only Insiders is installed.
• Add a unit test asserting expected LLMBinaryName values for LLM-gateway-capable client configs.

Reviewed changes

Copilot reviewed 2 out of 2 changed files in this pull request and generated 2 comments.

File	Description
pkg/client/config.go	Corrects the VS Code Insiders binary name used during LLM gateway client detection.
pkg/client/llm_gateway_test.go	Adds a regression test validating real supportedClientIntegrations LLMBinaryName values.

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[codecov]

Codecov Report

✅ All modified and coverable lines are covered by tests.
✅ Project coverage is 67.03%. Comparing base (9c66743) to head (3b1345d).
⚠️ Report is 1 commits behind head on main.

Additional details and impacted files

@@            Coverage Diff             @@
##             main    #5109      +/-   ##
==========================================
- Coverage   67.08%   67.03%   -0.05%     
==========================================
  Files         595      595              
  Lines       60000    60000              
==========================================
- Hits        40250    40222      -28     
- Misses      16691    16720      +29     
+ Partials     3059     3058       -1     

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