There is a signifigant difference between raikiri and chidori.
Chidori is the technique kid kakashi and sasuke learned. It is simply generating lightning around the hand, and useing it to cut/stab.
Rakiri, is the evolved technique of chidori. It is almost identical, but is stronger. Rakiri, the user forms more of a orb, and a greater concetration of lightining around the hand.
If kakashi used raikiri vs chidori, his raikiri would win.
Now since the differences are so minor, it really doesnt matter. Raikiri cannot be used as a name for chidori, but chidori can still be used for a raikiri user.
Again, there is a difference(other than name) between raikiri and chidori. But it is very minor.
An example of a different jutsu with a very similar predicament.
You cannot call a rasengan "oodoma rasengan", but you can call a oodoma rasengan just "rasengan".
Saying oodoma isnt any stronger than rasengan, would not be accurate. However for the most part, they are still the same.