What is an Undeclared War?

Following the XYZ affair, America and France took part in an "undeclared war" called the Quasi-War. After hearing this in the documentary we watched in class, I was interested in what exactly an undeclared war was, and if there had been others in US history that affected the nation just like a war that was "declared".
In my research, I found that the United States has not officially declared war since World War 2, even though it is plain to see that the government has engaged in various disputes worldwide, including but not limited to the conflict in Syria and the Middle East, the Vietnam War, and the never-ending 'war on terror' that began after 9/11. The way that they got around the legality of engaging in warfare without declaring war is part of The War Powers Resolution of 1973 as well as congressional Authorization for Use of Military Force or AUMF.
The War Powers Resolution of 1973 established that there were two ways for the armed forces to be deployed: either a declaration of war or in the case of a "national emergency created by an attack upon the United States, its territories . . . or its armed forces.". This meant that if the government interpreted a situation to be threatening to the United States, they could utilize the armed forces without officially declaring war, which would be a much more difficult diplomatic situation.
Consecutively, the government could also engage in war with a congressional Authorization for Use of Military Force or AUMF. AUMF was created directly in response to the September 11th attacks and states that the government can take action against any person, organization, or group that could have taken part in the planning or execution of said attacks. Much like the loose interpretations of the constitution that were a source of dissension between political parties in early United States history, this loophole created a way for the government authorities to attack countries and not label it war, instead, implying that it was counterterrorism. 
In the history of the United States, there has not been a lot of formally declared war in many years. However, there has been a lot of conflicts, so even though it may seem like there is a lot going on, the government is able to avoid jumping through the hoops of declaring war and taking strong stands by utilizing the many loopholes they created for themselves over the course of history.

https://www.smithsonianmag.com/smart-news/unremembered-us-france-quasi-war-shaped-early-americas-foreign-relations-180963862/
https://www.senate.gov/pagelayout/history/h_multi_sections_and_teasers/WarDeclarationsbyCongress.htm
https://nationalinterest.org/feature/america-addicted-fighting-undeclared-wars-20535
https://www.thenation.com/article/american-warfares-giant-open-secret/