England & Wales unpaid invoice: is a statutory demand the same as a small claim?

No. A statutory demand is an insolvency-related demand, not simply another name for a county court money claim. GOV.UK says a debtor normally has 21 days to pay or agree payment after receiving a statutory demand, and different consequences can follow. For smaller debts, GOV.UK warns there may be faster ways to recover money. Official source: GOV.UK Statutory demands. Guide: https://getinstantlaw.com/small-business/customer-wont-pay-invoice