45 Colt is the correct name. There was never a '45 short Colt' round as there were with other calibers which lead to the short/long names.

There were shorter 45 rounds at the time (curiously even a round that was shorter than the original 45 Colt, but was also named 45 Colt), which is where the name 'long colt' came from, but Colt never branded the round anything but 45 Colt. I have heard of boxes straight from Colt labeled '45 Long Colt', but have never seen one nor seen a picture of one with one exception.

Colt made a wax bullet set which they did indeed put '45 long colt' on the box. If you count that, then yes, Colt has designated it '45 Long Colt'. I don't count the wax bullet, and in a regular round, until I see a picture of the mythical box of Colt made '45 Long Colt' rounds, I'll stick with Colt's official stance on the subject, that it's never been anything but '45 Colt'.

But I don't get my undies in a twist over it either. Saying you want a box of 45 long colt won't confuse anyone, so I don't really care.