Scientific name for the problem given in the description. (In machine learning what kind of problem it is termed as)
Ok. I think I'm getting a clearer picture. Firstly any name isn't going to be different just because the data is large. Of course in the big data community anyone is free to coin a term and time will show if it gets adopted (same goes with what we might consider a scientific name for that matter too).

Another thing is input output. These terms can be confusing as what is input by a comp scientist into a training model is both the input and output of what a statistician would define.

I'm not aware of a name for the problems but names for models addressing such a problem include logistic regression and probit regression. Both deal with a(n) (set of) independent variable(s) - the feature name in your case - and a resultant dependent variable - the binary relevancy. The regression looks at fitting a linear function to a hidden variable; whose threshold gives the binary result. Fitting an "input" training set then allows the scoring based on this linear function, whether you want a binary score or a real valued one (from the hidden variable) that will be more indicative of the probability of your prediction.