Yes, the general consensus is that the war ended after the signing of the Korean Armistice Agreement.
Both countries signed the armistice, thereby ending the war in a truce.
The Korean War, which began on June 25, 1950, when the North Koreans invaded South Korea, officially ended on July 27, 1953. At 10 a.m., in Panmunjom, scarcely acknowledging each other, U.S. Army Lt. Gen. William K. Harrison, Jr., senior delegate, United Nations Command Delegation; North Korean Gen. Nam Il, senior delegate, Delegation of the Korean People's Army and the Chinese People's Volunteers, signed 18 official copies of the tri-language Korean Armistice Agreement.
However, this Armistice Agreement was only meant to stop hostilities and was not a permanent peace treaty between both nations. Excerpt from the treaty below:
will insure a complete cessation of hostilities and of all acts of armed force in Korea until a final peaceful settlement is achieved
As such, the Korean War can be described as a "Frozen Conflict", as quoted from Wikipedia:
a frozen conflict is a situation in which active armed conflict has been brought to an end, but no peace treaty or other political framework resolves the conflict to the satisfaction of the combatants. Therefore, legally the conflict can start again at any moment, creating an environment of insecurity and instability.
So, simply put, the armistice ended the Korean War, however, it was meant to be a temporary solution. Since both sides could not agree to a peace treaty, it is a general consensus that the war ended in 1953 after the signing of the armistice.
It's worth noting that the CNN article that you cited is an opinion piece which reflects the opinion of the author, not the actual status of the war.