demo: rename auth route to /api/auth/token and fix stale JSDoc

[umair-ably]

Summary

Demo-side of the AI Transport auth consistency work. Companion to the docs PR at ably/docs#3397.

The endpoint returns a JWT, but its path read as Ably's native Token type. That's the exact JWT-vs-Token confusion the docs convention is now killing. This PR brings both demos onto the agreed neutral path.

What changed

• Route directory renamed: src/app/api/auth/ably-token/ → src/app/api/auth/token/ in both demos (use-chat, use-client-session). Done with git mv so history follows.
• Client fetch updated in providers.tsx (both demos) to match the new path.
• Stale JSDoc fixed in route.ts: the comment claimed the client connects with authUrl, but the code uses authCallback. Reworded to match the code.
• See: link repointed from the now-redirected sessions-identity/identifying-users-and-agents URL to the canonical concepts/authentication page.
• README + docs (docs/get-started/vercel-use-chat.md) updated to use the new path, so the documented setup matches the running demo code.

Why /api/auth/token

Type-neutral on purpose. The endpoint returns a JWT — a path containing ably-token reads as the native Ably Token type and was actively misleading. /api/auth/token carries no token-type bias.

Out of scope

• Production-hardening of the auth server (clientId should be derived from an authenticated session rather than trusted from the request). The demos remain demo-simple by intent. The docs PR carries the security framing for production callers.

Test plan

• CI green
• Both demos start (pnpm dev) and the chat flow round-trips successfully against the new route
• grep -rn "ably-token\|sessions-identity" across the repo returns no hits
• No regressions in pnpm run format:check

Note: pnpm run lint and pnpm run typecheck fail on test/helper/test-setup.ts in this local clone because the ably-common submodule isn't initialised. That failure is unrelated to this branch — the file is unchanged here. CI will run with submodules initialised.

🤖 Generated with Claude Code

[github-actions]

Coverage Report

Status	Category	Percentage	Covered / Total
🔵	Lines	92.96%	2830 / 3044
🔵	Statements	90.92%	3037 / 3340
🔵	Functions	92.62%	465 / 502
🔵	Branches	78.57%	1533 / 1951

File Coverage

No changed files found.

Generated in workflow #567 for commit ab92e74 by the Vitest Coverage Report Action