1) click "add a new address" (if you use "update" it updates an existing address record which might be the same for shipping and billing at this time)
2) fill all fields for the new address record, and give it a name/alias, like "mom, sister, home, work etc.."
3) back in the checkout / addresses section, uncheck the checkbox at "Use the delivery address as the billing address"
4) choose the address from the dropdowns "choose a delivery address" and "choose a billing address"
Alternatively you can watch a video here how this is supposed to be done:
This video shows how you can add more addresses to your account and select different addresses for shipping and billing.
Link to the Video on YouTube: https://youtu.be/6XTytqyvlhM