refactor(auth): single-source Better Auth config + split package exports (#45)

[pranavp10]

Summary

• Collapses the two divergent betterAuth() configs into one runtime instance in @reloop/auth/server (auth service becomes a thin re-export).
• Adds @reloop/auth/types and rewires the browser client so it derives types from the single source without value-importing server deps.
• Moves signup-invite helpers + Redis secondary storage into the shared package; keeps client/types isolation under test and CI.

Closes #45

Test plan

• bun run test in packages/auth (export isolation — 2 pass)
• bun run test in apps/backend/auth (characterization tripwire — 6 pass)
• bun run typecheck / check-types for @reloop/auth and be-auth
• tsc --noEmit for dashboard, console, docs, links
• CI BE Auth Docker workflow green (isolation + tripwire gate image build)

Greptile Summary

This PR centralizes the auth package around one Better Auth server runtime. The main changes are:

• Shared server config and runtime exports under @reloop/auth/server.
• Backend auth service files now re-export or consume the shared package implementation.
• Signup-invite helpers and Redis storage moved into packages/auth.
• New @reloop/auth/types export and client import rewiring.
• Export-isolation tests and CI coverage for the shared auth package.

Confidence Score: 4/5

This is close, but the type export issue should be fixed before merging.

• @reloop/auth/types still points TypeScript at the server runtime implementation.
• Browser packages can still be forced to resolve server-only dependencies during type checking.

packages/auth/src/types.ts

Important Files Changed

Filename	Overview
packages/auth/src/types.ts	Adds the public type surface, but it still derives exported types from the server runtime implementation.
packages/auth/src/client.ts	Updates the client to consume the new type export while keeping runtime imports client-side.
packages/auth/src/server/auth.ts	Moves the Better Auth runtime into the shared server package entrypoint.
packages/auth/package.json	Adds split package exports for server config and auth types.

_{Reviews (2): Last reviewed commit: "fix(auth): address PR #54 review finding..." | Re-trigger Greptile}

Greptile also left 1 inline comment on this PR.

[greptile-apps]

Type Surface Still Resolves Server

import type keeps this out of the emitted bundle, but TypeScript still has to resolve ./server/auth to compute typeof auth. That module imports the server runtime graph at the top level, including DB, bus, Redis, and Better Auth adapter modules. A browser package that imports User or Session from @reloop/auth/types can still fail type checking or package resolution unless those server-only dependencies are also resolvable. Derive these exported types from a declaration-only contract instead of the runtime implementation module.