Well I dont know how she is in XIII but saying this only from duodecim she didnt seem that cold. She ended up opening herself and all but she was a bitch to laguna xD. Well She kind of developed a liking for kain and showed some selflessness. So I guess she is not a bad character.. And I dont see where are the similarities to Cloud taking into consideration only 012.
She's pretty mellow in duodecim. I can tolerate her in there. But thats about it
I think it's unfair to base a character on Dissidia - It was just a compilation game, not a canon series title nor where the characters in their storylines.
As for Lightning, This is a really tricky one as you have to divide her from the poor executed story form the game. I generally have no problem with her in Final Fantasy XIII-1, Yes she was cold at first, but so was Squall and I regard him in my top 5. She does express herself well through out the game and you see her feelings develop. But unfortunately a lot of this feels separated from the plot which was presented in such a sloppy manner. I should also state that I refused to read any of the novellas or datalogs for extra information. You shuoldn't have to as the game has enough cutscenes, the story and depth should be presented in the game.
As for Lightning in XIII-2. The problem to me is that you rarely see her, She is not even in your party so her presence is hard to feel. I understand that you're meant to see her though Serah's eyes, but It just felt more distant to me. I would have rather continued as Lightning than played as Serah (All though I did like Noel and his conflict with Caius.) - So becuase of XIII-2 I do feel quite distant from her now. Will my mind be changed when she comes back into the foreground for this new game... I don't know... I was sorting to hoping to controlling a Team of Lightning. Hope and Noel in an XIII-3 game, but as we all know this new title is something else...
She comes around rather easily. But its like no personal development uappened in the game compared to the rest of the characters. The only ones i think were great were Hope and Sazh. And the very reason of being similar to cloud and squall is why people hate her. She's not original.