JeremyFlight

[spice pewter]

If I were to use a paymentElement instead, would this include fields for the cardholder's name and postcode, as well as the usual card number, exp date and CV2 number?
Yes, you can use Stripe Element for this. You can check the preview:
https://stripe.com/docs/payments/quickstart

[steady oar]

So I would do something like this?

var paymentElement = elements.create('payment');

[spice pewter]

not just that, you need to follow that quickstart step by step.