Centralize seat capacity accounting and the payment provider registry.

Replace manualProviders.ts with a paymentProviders.ts registry
(automatic vs manual settlement) and move all seat counting into
capacity.ts as the single source of truth: only paid/checked-in tickets
and pending_approval payments hold a seat, so abandoned checkouts never
block sales. Admins can now knowingly approve a payment over capacity
(allowOverCapacity), with the booking and admin UIs updated to match.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

backend/src/lib/paymentProviders.ts

@@ -0,0 +1,37 @@
+// Payment provider registry.
+//
+// Every provider is either:
+//   - 'automatic': the gateway itself confirms the payment (webhook/invoice
+//     settlement) and the booking is auto-approved on success. No admin involved.
+//     Currently Lightning; future online gateways (e.g. Stripe) go here.
+//   - 'manual': a human must verify the money arrived (TPago, bank transfer,
+//     card handled offline, cash at the door). These are never auto-confirmed
+//     and never auto-failed; an admin settles them by hand. Bank transfer and
+//     TPago additionally expose an online "I've paid" step that moves the
+//     payment to 'pending_approval'.
+//
+// Capacity note (see lib/capacity.ts): only paid/checked-in tickets and
+// 'pending_approval' payments hold a seat. A bare 'pending' payment — of either
+// kind — holds no seat, so an abandoned checkout can never block sales.
+
+export type PaymentProviderKind = 'automatic' | 'manual';
+
+export const PAYMENT_PROVIDERS: Record<string, { kind: PaymentProviderKind }> = {
+  lightning: { kind: 'automatic' },
+  tpago: { kind: 'manual' },
+  bank_transfer: { kind: 'manual' },
+  card: { kind: 'manual' },
+  cash: { kind: 'manual' },
+};
+
+export const MANUAL_PAYMENT_PROVIDERS = Object.keys(PAYMENT_PROVIDERS).filter(
+  (p) => PAYMENT_PROVIDERS[p].kind === 'manual'
+);
+
+export function isManualProvider(provider: string): boolean {
+  return PAYMENT_PROVIDERS[provider]?.kind === 'manual';
+}
+
+export function isAutomaticProvider(provider: string): boolean {
+  return PAYMENT_PROVIDERS[provider]?.kind === 'automatic';
+}