Anyone who has been learning for a while will have realized that for different halachos there are different definitions of the word “day”. In general in the Torah the word “day” means either the daylight hours, or a period from one nightfall to the next. In kodshim, the word day means a night following a day as opposed to preceding a day. Furthermore, the Rav zt”l pointed out (what was really an explicit Riva at the beginning of the fifth perk of Yoma) that the time between sunset and the appearance of three stars might not be a sofek in the sense of us being unable to clarify whether it is day or night. It might really be both and the sofek is which Halacha, that of day or that of night, should we apply. However, the perception of a day in the Tosafos on the daf today is a little different (although perhaps more commonsense).
The Gemara at the end of the second perek of Nazir discusses the opinion of Rabbi Yossi that if a zav or zavah is about to become tahor on the night of Peach (as in they are on the final day of the count and they are within the cases that do not require the bringing of a korban and they have already gone to mikveh) then a Korban Pesach can be slaughtered for them and it’s blood sprayed on the Mizbeach. If they then see either more blood or another seminal emission causing them to become impure again and preventing them from eating the Korban Pesach they need not bring the Korban Pesach again during Pesach Sheini even though they did not eat the korban (which we assume is the primary obligation). Accordingly, the Gemara assumes that Rabbi Yossi must hold that the fact that they saw more blood or another emission does not render them retroactively impure for the previous portion of the day succeeding when they went to the mikveh. The Gemara then asks that if they are capable of becoming definitively pure between the times that they see either blood or a seminal emission, how is it that they can have three successive emissions such that they would become impure for a full seven days or be required to bring a korban. The Gemara answers that it must be that Rabbi Yossi will only apply this law if a woman either saw blood continuously for three days or either a man had seminal emissions or a woman saw blood immediately before sunset for three days successively preventing the beginning of a day with the possibility of going to the mikveh. Even if they were to go to the mikveh that day it would not be a “clean” day because they had already seen something so even post mikveh the count could not continue. Rashi in Pesachim 81a ruled based on this that if the night started without seeing either blood or a seminal emission, it would be permitted to count the following day as one of the clean days (as long as nothing was seen for the rest of the day). Tosafos here disagree. They state that the beginning of the night cannot allow us to count the following day, although they seem to imply that it counts (where it is relevant) to allow us to count the previous day as a clean day if we do not need the entire day to be clean. Accordingly, it seems that Tosafos are defining a day as the period we think of as a day – from when we wake up until a bit into nighttime for the purposes of this halacha. For this halacha what matters more is not the formal definition of a day but the experience of a day.
If so, the problem the Gemara had with Rabbi Yossi’s opinion was not that you could count the start of a day as pure and thus prevent the later sighting from being added to the previous sighting . The problem the Gemara had was that the successive part of the day after a sighting and after a person went to mikveh could allow that day to be counted as a clean day (except in cases where you need a full clean day such as zavah gedolah) and accordingly, the answer the Gemara gave was that the sighting must have happened at the end of the day so there would be no time before nightfall in which to consider the day clean, not that the sighting happened the previous day right before nightfall so that the beginning of the night was not clean.