Create the PostHog project and expose its key to web/ and server/

[georgekaran]

Context

Both #24 (frontend events) and #25 (server-side events) need a PostHog project to send to. It was originally bundled into #24's scope simply because #24 was the first issue to need it — which made #25 depend on a frontend ticket for something that is not frontend work.

Pulling it out here so either side can proceed independently. This is a signup and some environment config, not a feature.

Scope

In

• Create the PostHog project (one project, both clients).
• Project API key available to web/ and to server/ — same key, separate env vars, since Vite inlines its own at build time and must never read the server's environment.
• Region choice recorded in DECISIONS.md (EU vs US changes the ingest host and is annoying to move later).
• Confirm the ingest host is set explicitly rather than left to the SDK default, so the two clients cannot end up pointed at different regions.

Out

• Any capture() calls. Frontend events are Product analytics: PostHog on the frontend #24, server events are Product analytics: server-side events #25.
• Flags and surveys (Feature flags and in-app survey #26), which use the same project but need no extra setup here.

Notes

The key that ships to web/ is public by design — posthog-js exposes it in the bundle, and that is expected. It is not a secret and does not need secret handling; it just needs to not be confused with a personal API key, which is one and is only needed if #26 later does local flag evaluation.

Acceptance criteria

• PostHog project exists; region recorded in DECISIONS.md.
• Key readable from web/ and from server/ via separate env vars, documented in .env.example.
• Ingest host set explicitly in both clients' config.