chore(chapter-08): give the API workspace its own project config

Chapter: 08 - Workspace independence.

Plain-English map:
Add the API package metadata, TypeScript config, and Vitest config needed for
`apps/api` to build, run, and test as a clear workspace.

Why it matters:
After the monorepo move, each workspace needs its own local controls. A new
contributor should be able to enter `apps/api` and understand its commands.

Principle:
Every package should be independently understandable even inside a monorepo.

Precedent borrowed:
Monorepo conventions used by larger TypeScript open-source projects and by
forks that split Mike into clearer package boundaries.

Upstream base: willchen96/mike@d39f580.
Original local commit: 377d685.