Will my domain name be registered under my name?
Absolutely! You will be paying for the domain, so you will have complete ownership.
You can lose control over your domain in the event of non-payment. If you fail to pay, Crucial will retain control of the domain—including the right to transfer, sell, or modify it—until your balance is paid. This is nothing unique to Crucial, this is how all registrars operate.
Even if you stop hosting with us, so long as you pay your yearly registration fee of $5.99, your domain is 100% under your control.