I would like to delete my account completely and not appear registered in Dynu.

You can go into the 'My Account' area and remove your account completely.

BTW, you must first remove all dynamic DNS hosts first before trying to remove your account. This is a fail safe to ensure, you don't lose any hosts without complete knowledge.

I'm in the same position - but there's no option to delete the account from the "My Account" section. Is there a new option?

You need to make sure that the hostnames are removed from the DDNS Services and you do not have any other paid services in your account. If you do, you may need to contact customer service.