The story you are about to see is true. The names were not changed to protect the innocent until the show went to color in the next decade. Or if they were changed, it wasn't mentioned.
This episode from the original black & white (as seen above in three videos) was refilmed with the exact same script and almost all the same actors in the late 1960's in color, also co-starring Barry Williams who later played Greg on The Brady Bunch.
It is, if I'm not mistaken, Dragnet's only Christmas show and was titled BIG LITTLE JESUS.
Isn't it sad that we have to go back to the 50's and 60's to find a show on TV that displays any reverence to God or to the Catholic Church? I can hardly think of any today that don't portray Christians as odd-balls or portray priests as corrupt or perverse. The Closer might be one, possibly.