Initially they referred to Kakashi's move as Lighting Cutter (Raikiri), but then they stopped calling it that and called it Chidori from then on. Any specific reason for this? Seems really odd to change a move's name after establishing it as something else.
The magic of wiki:
The Lightning Cutter is an enhanced and concentrated form of the Chidori with the same effects and drawbacks. According to Might Guy, it gained its name after Kakashi split a bolt of lightning with it. Because the Chidori is already powerful on its own, the Lightning Cutter requires better chakra control, which is exemplified in its appearance. The Chidori appears as simply a mass of white electrical chakra in the user's hand, while the Lightning Cutter is more focused (and blue in the anime).
tl;dr: It's still a Chidori but when Kakashi's using it, it's more refined and famed for cutting a lightning bolt ergo dubbed Raikiri.
Originally the name of the jutsu is Raikiri (Lightning Cutter). However, because the jutsu makes a noise like that of a thousand bird squeaks, it was dubbed by others as Chidori (千鳥) which literary means a thousand birds.
AFAIK Chidori and Raikiri is actually the same jutsu. The name is different due to the reason stated above. Also, Kakashi prefer to call it Raikiri while Sasuke, who was taught by him, prefer to call it Chidori.