feat(payments): add payments module with embedded checkout

[naor-gal]

Summary

• Adds base44.payments (browser) with completeCheckout(), listPurchases() and mountCheckout().
• Adds base44.asServiceRole.payments (backend functions) with createCheckout(), completeCheckout(), listPurchases(), refund(), cancelSubscription(), pauseSubscription() and resumeSubscription().
• The service-only methods aren't on the user module type, so calling them from the browser fails at compile time. This follows the ConnectorsModule / UserConnectorsModule split.
• Request and response fields use the API's snake_case names unchanged, like the other modules. Failed requests reject with the existing Base44Error, with code from the API's error envelope.
• mountCheckout() renders an embedded checkout in any framework, with no new dependencies:
  • PayPal: loads PayPal JS SDK v6 for the checkout's environment and renders a <paypal-button>. On approve it calls completeCheckout(). A declined payment goes to onError, and the button stays so the buyer can re-approve the same order.
  • Stripe: loads Stripe.js v3 and mounts Embedded Checkout. Mounts run in order, and a new one replaces the previous one, since Stripe allows one per page.
• Docs: JSDoc for every public type and method, a provider support table, and the error codes, purchase statuses and recurring frequencies on the payments reference page. A separate service-role-payments page holds the service-role methods. Also adds a README entry.

Notes

• Base44Client gains a required payments member and asServiceRole gains payments, the same as earlier module additions. Code that builds that type by hand needs the new members.
• The public element type is the structural CheckoutElement, so the SDK's types don't need the DOM library in backend code.

Test plan

• npm run lint
• npm test -w @base44/sdk: 362 tests, including new unit tests (HTTP routes, error envelope, mount lifecycle) and type tests
• npm run build -w @base44/sdk
• npm run create-docs (packages/sdk): payments and service-role-payments pages generate cleanly
• Manual, PayPal sandbox: created an embedded checkout, mounted it, approved as a sandbox buyer, and completeCheckout() returned paid
• Manual, PayPal sandbox: re-approved the same order from the mounted button, and capture succeeded
• Reviewer: spot-check the generated payments.mdx page in mintlify-docs

🤖 Generated with Claude Code

[github-actions]

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