Reality/causality also moves at the speed of light, and observations of time and space are relative to the observer. So it's still present reality, for you at least.
Ah but that is just nerve sensor input time, it does not include the time spent processing inputs into an awareness (or however one wants to describe our internal model.) Perceptual lag is a subject of a lot of unfortunately paywalled research.
The lag is not dissolved by relativity. Reality, as many athletes know, will let you know when your perceptual lag is a critical shortcoming.
I suspect perceptual lag is the term. I had heard it from a memory standpoint, I think. Something like:
Our minds operate on our memories, our memories are always beind reality. Our interpretation affects our memories, or at least the recall, and are always reacting to the past.
Perceptual lag seems more to the point. (Long thing included in case someone else happens to have another idea.)