chore(chapter-09): move API source into the workspace path

Chapter: 09 - Physical source move.

Plain-English map:
Move the API implementation files from the old backend location into
`apps/api` so the directory structure and package boundaries match.

Why it matters:
The repo should not say one thing in package metadata and another thing in its
file layout. Matching structure reduces mental bookkeeping for readers.

Principle:
Make ownership visible in the filesystem.

Precedent borrowed:
The same workspace and modularization pattern introduced in Chapter 01.

Upstream base: willchen96/mike@d39f580.
Original local commit: d1ec379.