Seems to be standard terminology for time series classification to me, to be honest. I think the approach would also work if there are duplicates in the data. Although the estimate would be overly optimistic, right?

With their notation they have not specified that the T's are unique. So, a first fix up would be just to state that the T's were distinct. And it would help to be explicit that i from 0, 1, 2, ... corresponded to increasing time. Moreover, is the data equally spaced in time? Likely, yes, and in that case, clearly say so.

No, i indexes the patient, not time. (T_0, y_0) is one patients entire time series.