Rob
I suspect you have done everything correctly and there is no problem with the documents. Sometimes when you arrange cover without any No Claims discount, Insurers will offer an introductory discount. This is often shown as 1 year on your documents. I have known this to be the case with other Insurers and expect it is the same with Admiral
Contact the Customer Relations or Quality Team, whatever they call it. Their telephone no. is 0800 952 1300
Make a complaint about the service you received when you phoned. The person should have been more professional and explained the process properly. Many staff in these call centres do not have a clue about Internet processes and associated documents.
In regard to any admin fee for cancelling within 14 day cooling off period, I think this is dodgy. If the policy does not meet your requirements, you have the right to cancel, without any charges.
I am pretty sure when you call their complaint handlers you will receive correct information and an apology. Don't be afraid to ask for compensation for the hassle you have encountered. These complaint teams usually have store vouchers available to them. Ask for £50 voucher of the store of your choice. They will find it difficult not to offer something. If you don't ask you don't get!!
Hope all goes well
Huckster