The bar says the shopper has reached free shipping, but checkout disagrees

That should not happen, because the bar measures the cart exactly the way WooCommerce's own free shipping method measures it — the same subtotal, the same handling of tax display, and coupons subtracted only when the method is set to subtract them.

It reads the same Free shipping method checkout uses, too: the one in the shopper's own shipping zone. A free-shipping coupon counts only where that method is set to accept one, and a method that needs A minimum order amount AND coupon is only called unlocked once the coupon is applied as well.

If you see a disagreement, check the Setup screen: if Coupons and discounts has been changed away from Follow the shipping method, the bar is deliberately counting differently from checkout. Setting it back to Follow the shipping method restores the guarantee.