If I pay online with internet banking, that is I pay for a set of tyres but use my credit card, will this be charged as a cash withdrawal or a charge for tyres?
It will be charged as a cash withdrawal. Additionally, you will incur interest immediately at the cash rate and not receive any rewards (If your card offers rewards).
I know if the Merchant is Registered to accept Bill Payments and they are happy to wear the Merchant Service Fee then this could go as a Payment otherwise as Michaelnz advises, this will be a cash withdrawal. It all comes down to the Merchant and there agreement with BNZ.
Here is a link of the list of Companies Registered for Credit Card Payments.
I hope this is of help.