Yeah, it’s definitely Kylo’s saber unmodded.

Though I like to think that Ben possibly used it before building his own saber. When Finn uses it against Kylo in the forest, Kylo clearly recognizes it, since he says “It belongs to me.”
I think he said it because he sees it as his birthright, but since he seems familiar with it, I wouldn’t be surprised if he had used it before. Just like Luke used it before building his own.

If that’s the case, then I could add some more context to why Luke just tossed the saber away in TLJ.

I think if you can make that change Nev, make it, and people can make their own assumptions about it.