[codex] Add PostHog GTM analytics

[blove]

Summary

• Add PostHog browser initialization and typed website analytics helpers.
• Capture GTM interactions across nav, footer, docs, whitepaper, newsletter, and lead flows.
• Add server-side conversion capture for lead, newsletter, and whitepaper API routes with PII-conscious properties.

Validation

• npx vitest apps/website/src/lib/analytics/properties.spec.ts --run
• npx nx build website
• npx nx e2e website
• git diff --check

Deployment Notes

• GitHub repo values are configured for NEXT_PUBLIC_POSTHOG_HOST and NEXT_PUBLIC_POSTHOG_CAPTURE_LOCAL.
• GitHub repo secret NEXT_PUBLIC_POSTHOG_TOKEN is configured.
• Vercel production env still needs the same values, or the workflow needs to pass GitHub env into the Vercel build.

[vercel]

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Project	Deployment	Actions	Updated (UTC)
cacheplane	Building	Preview, Comment	May 2, 2026 5:07pm