Make examples reproducible

[sueszli]

Refactored examples to improve reproducibility.

The existing example_app could not be run due to an Postgres failed to become healthy error and was moved to example/web-app.

Introduced a minimal Postgres bootstrap (~5 LoC, based on https://gist.github.com/sueszli/4165d5e388cf289978ea64fdcda7661b) and added example/demo/demo.py, which supports both a real Postgres instance and the in-memory variant.

Additionally, reproduced the README example as a self-contained uv-inline version in example/tiny-demo.

[piercefreeman]

If the goal with this PR is to be able to run our example app programmatically, it feels like it has a bit too much overlap with our existing testing infrastructure:

https://github.com/piercefreeman/waymark/blob/main/example_app/Makefile#L21

I would want to make sure we harmonize these and aren't duplicating workflows or pipelines to run this logic.

[piercefreeman]

Right now this seems to duplicate the code in workflow.py - is it intended the be the same or different? If the same, we should consolidate them into the same place.

[sueszli]

is it intended the be the same or different?

This version replaces the entire example_app with a minimal backend-only example without a seperate Dockerfile, docker-compose, Makefile, uv.lock, pyproject.toml, .python-version file.

The script itself can either:

• Cover the same functionality: https://github.com/sueszli/waymark/blob/e2541c335ea9891d17b8e5303b62227713317474/example/demo/demo.py
• Or the one from the README: https://gist.github.com/sueszli/f6ad8ae9631e4652ef403859b4c4d1cd

[sueszli]

Closing. It's smarter to consolidate into the same place and patch the Makefile first.

Opened new PR: #170

[sueszli]

Just noticed the behavior here is completely different because we're pulling from PIP and not the latest wheels.