[skills] Refresh the AI Gateway skill

[jerilynzheng]

Summary

• Rewrites skills/ai-gateway around current sources: live model discovery, API key and OIDC authentication, compatible clients, routing, budgets, reporting, Logs, and coding-agent setup.
• Removes stale and unsupported claims: static model defaults, AI SDK v6-era imports, 24h OIDC lifetime, whole-response cacheControl, per-user rate-limit settings, Slack alert channels, and a non-existent logs API.
• Moves details into focused references for setup, routing, spend/observability, and coding agents, following the same reference split as the OpenRouter skill pattern.
• Refreshes injection metadata: prompt signals, current docs URLs, narrower model-slug validation, provider-key BYOK handling, and chaining to the ai-sdk skill for SDK implementation.

Validation

• bun run validate passes.
• bun run typecheck and bun test pass: 974 tests, 0 failures.
• All build checks pass: build:skills, build:manifest, and build:from-skills are up to date.
• Every referenced model slug was checked against https://ai-gateway.vercel.sh/v1/models; both model-list URLs return 200.
• Coding-agent support and flags were checked against the installed Vercel CLI 59.7.0 rather than the stale skill body.

Notes

• [skills] Compress ai-gateway skill body 54% (19.8KB -> 9.1KB) #145 is approved but dirty and now conflicts with this refresh. This PR uses its compression direction where current, but replaces the stale guidance rather than only shrinking it.
• The docs PR in vercel/front#83733 can point to this skill after merge.