Is the PDF invoice in the order confirmation email the same as my invoice template?

No, they are separate. The PDF attached to the customer's order confirmation email uses a standard Wcart invoice layout. It is made automatically from the order's details and doesn't use your invoice templates.

Your templates under Settings → Store Settings → Invoice Templates control the invoice you print from an order with Print invoice.

If you want customers to have your designed invoice, open the order, click Print invoice, and save it as a PDF from the print window. Then send it to the customer.