Civil war is that which is waged between two parties, citizens or members of the same state or nation. National war is a contest between two or more independent nations, carried on by authority of their respective governments.
The Confederacy had a government formed in the same fashion as the union government, comprised of independent states ratifying a new Constitution. In the Prize Cases, Justice Robert C. Grier noted that the right to blockade Confederate ports was 'jure belli', from the law of war and peace [aka international law], not from any power enumerated within the US Constitution. Additionally, instead of noting that secession was illegal, he noted that the Confederate states 'right to do so is now being decided by wager of battle'.
The confederacy was never a legitimate country. Their secession was illegal. They lost war. It was a Civil War. The definition you use proves it.