I can't find anything from Into Darkness that states that there has not been any war with the Klingons up to this point.
I dunno whether DiscoTrek can really be considered a Kelvin prequel. I like the idea, since DT clearly draws so much influence from the Kelvin timeline - it seems like it wants to be part of it. But I'd have to sit down and look at the dates and evidence to see if that idea really worked.
In the KT the Enterprise's maiden voyage hasn't happened yet when Discovery takes place. In fact, it should be still under construction if Discovery was Kelvin Timeline.
Plus you know the completely different design.