You are almost correct. It is the Server Private Key would need to be generated with the ECDSA algorithm, as opposed to the commonly used RSA algorithm. That ECDSA Private Key could then be used to make a CSR, which in turn would be signed and made into the ECDSA Public Key.
This particular cipher suite, ECDHE-ECDSA-AES256-SHA384, would generally be used in a dual-key environment where clients could fall back to an RSA cipher if they did not yet have support for ECDSA. If you used only ECDSA you would likely restrict many clients from reaching your server, as the key algorithm is not widely used. However, it's faster and will be utilized more in the future.