regarded themselves

You have kind of answered your own question in a sense. Calling oneself a country does not a country make. The very act of seceding from the Union was seen by many in the North, particularly Lincoln, as an illegal act, and one which needed to be resolved using diplomacy if practicable, but by force if necessary. Lincoln maintained throughout the war that the Southern states (or any states) had no right to secede, and stated so on several occasions. In his inaugural address, Lincoln laid out his argument thus: (All bold text mine)

If the United States be not a government proper, but an association of States in the nature of contract merely, can it, as a contract, be peaceably unmade by less than all the parties who made it? One party to a contract may violate it--break it, so to speak--but does it not require all to lawfully rescind it?

Descending from these general principles, we find the proposition that in legal contemplation the Union is perpetual confirmed by the history of the Union itself. The Union is much older than the Constitution. It was formed, in fact, by the Articles of Association in 1774. It was matured and continued by the Declaration of Independence in 1776. It was further matured, and the faith of all the then thirteen States expressly plighted and engaged that it should be perpetual, by the Articles of Confederation in 1778. And finally, in 1787, one of the declared objects for ordaining and establishing the Constitution was "to form a more perfect Union."

But if destruction of the Union by one or by a part only of the States be lawfully possible, the Union is less perfect than before the Constitution, having lost the vital element of perpetuity.

It follows from these views that no State upon its own mere motion can lawfully get out of the Union; that resolves and ordinances to that effect are legally void, and that acts of violence within any State or States against the authority of the United States are insurrectionary or revolutionary, according to circumstances.

I therefore consider that in view of the Constitution and the laws the Union is unbroken, and to the extent of my ability, I shall take care, as the Constitution itself expressly enjoins upon me, that the laws of the Union be faithfully executed in all the States. Doing this I deem to be only a simple duty on my part, and I shall perform it so far as practicable unless my rightful masters, the American people, shall withhold the requisite means or in some authoritative manner direct the contrary. I trust this will not be regarded as a menace, but only as the declared purpose of the Union that it will constitutionally defend and maintain itself.

In doing this there needs to be no bloodshed or violence, and there shall be none unless it be forced upon the national authority. The power confided to me will be used to hold, occupy, and possess the property and places belonging to the Government and to collect the duties and imposts; but beyond what may be necessary for these objects, there will be no invasion, no using of force against or among the people anywhere. Where hostility to the United States in any interior locality shall be so great and universal as to prevent competent resident citizens from holding the Federal offices, there will be no attempt to force obnoxious strangers among the people for that object. While the strict legal right may exist in the Government to enforce the exercise of these offices, the attempt to do so would be so irritating and so nearly impracticable withal that I deem it better to forego for the time the uses of such offices.

Essentially, Lincoln is stating that the Union is perpetual, and states cannot leave it on their own, and to attempt to secede is by itself an act against the government of the United States, and he will do all in his power to maintain the Union. He later stated:

In your hands, my dissatisfied fellow-countrymen, and not in mine, is the momentous issue of civil war. The Government will not assail you. You can have no conflict without being yourselves the aggressors. You have no oath registered in heaven to destroy the Government, while I shall have the most solemn one to "preserve, protect, and defend it."

Jefferson Davis, in January of 1861, following South Carolina's secession ordinance, wrote to former president Franklin Pierce:

When Lincoln comes in he will have but to continue in the path of his predecessor to inaugurate a a civil war and, leave a soi disant democratic administration responsible for the fact. Genl. Cushing was here last week and when we parted it seemed like taking a last leave of a Brother.

I leave immediately for Missi. and know not what may devolve upon me after my return. Civil war has only horror for me, but whatever circumstances demand shall be met as a duty and I trust be so discharged that you will not be ashamed of our former connection or cease to be my friend.

Even Davis, president of the Confederacy, regarded what was occurring as "civil war."

In the definition you gave, faction is a better term than region regarding the US Civil War, solely because thousands in the Confederate states did not support the Confederate government, just as thousands in upper South slave states like Kentucky supported the Union. So in 1860, a large faction of the nation attempted an illegal act, initiated armed conflict against the United States, and the United States fought to maintain the nation. Even for the Confederates, much of their actions stemmed from their belief that the US government was on the path to do something they felt was unconstitutional: end slavery. They decided to "jump ship" as it were, and build their constitution pretty much on the American one, but with concrete guarantees of slavery's preservation.

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Even if we allow the Confederacy winning the war, it would still be classed as a Civil War, primarily because the Confederacy was not recognized as a nation and would not have been until after they had won.