Short answer: When in doubt, trust in Rothman.
Long answer: It depends. You only want to include time where you are actually at risk of the outcome in your calculation of the denominator. Time where you aren't at risk (known as immortal person-time) should never be used in the calculation of a rate.
In your case, if the outcome of interest is the onset of a disease, then yes, the moment they have the outcome, they have it. You stop counting their time in the denominator, and they move to the numerator. Doing otherwise underestimates the incidence rate. The answer to what the other formula is doing is well...doing it wrong. Or more likely, doing it with the best data that is available - such as when all a study has is a number of counts per time and a number of person-time for an interval. But it should always be noted that this is a subtle underestimation.
Frequently, one might actually assign times when you only know an interval to try to address this. To use a hospital example:
We have 100 patients, who stay for 1 month. During that time, we get 5 cases.
The underestimation is 5 cases/100 person-months (400 person-weeks). We can however say that, in our experience, most infections occur within the first say, week of hospitalization. So we will assume that all cases occurred in that week. Now it's 5 cases/ 100 person-weeks. Or, if we have reason to believe they all got it in the 3rd week - say, we know the Jello in the cafeteria was contaminated with norovirus, we can assume that all cases occurred then, and now its 5 cases/300-person weeks. Most often, people just pick the middle of the interval for unknown circumstances. The technique you're seeing, where you use the most time possible in the denominator, is the lowest estimation of the rate you can have.