When Did The Catholic Church Get It's Name?

Based on my early bible study teachers and large nearly toxic doses of Chick Tracts, I learned at the impressionable age of 14 years old that the Catholic Church started sometime after Constantine "made a deal with the devil" to mix paganism with Christianity and labeled the now befouled and derailed religion "Catholic." This apparently occurred in the 4th century while the "True Christian Church" went underground.  Yes, I believed this and now I am kind of embarrassed to admit I was so gullible. I even went through Western Civilization class in college still holding to my uber fundamentalist nonsensical beliefs.
     Fast forward 31 years and I am sitting in my bedroom in a town called Emmaus, Pennsylvania reading a book by Steven Ray called Crossing the Tiber. He describes how the Church was first called Catholic in 110 AD in the writings of the Early Church fathers, ( I lived 45 years and never heard of any church fathers, except maybe for Augustine, but he was Protestant I was told.)
I was flabbergasted that this group of early believers that loved Jesus and were being persecuted for their faith were Catholics which = Christian. Any other splinter groups claiming to be Christian had to prove their succession from an apostle. If they had no succession, they were not Catholic, and thus the faith was safely kept pure from the many heretical sects springing up claiming to be Christian.

James Akin does a great short talk on this in the video below:

"Wherever Jesus Christ is, there is the Catholic Church"
(St Ignatius Letter to  Smyrna, 110 AD)