Hot into Cold - for he holds that the lower item wins [it will turn the item poured into it hot or cold like itself] and the hot water won't boil the cold [receiving] water, it will just warm it.
And not cold into hot - for the lower water will cook the incoming (upper) water, since the "bottom wins".
in a cup - since he needs it for drinking and he doesn't want it to get too hot, and also, it is a kli sheini.
He forbids it -even hot into cold.
A person may put[/pour] hot [water] into cold [water] - even in a kli rishon since hot water poured into cold water won't cook, as I will explain. Though cold water may not be poured into hot water even in a kli sheini, since Beis Shammai made a decree forbidding pouring it into a kli sheini so that one doesn't get mixed up and pour it into a kli rishon [which is forbidden by the Torah]. Beis Hillel who argue and permit pouring cold water into hot water, that is only in a kli sheini, as the gemara says: "when is this said? By a cup..." which implies that they agree that doing this into a kli rishon is forbidden. The logic here is not that "the bottom wins" as Rashi explains it, for if so this should be a question on the one who holds that "the top one wins". According to the one who holds "the bottom one wins" there is still a layer which does get cooked as I will explain later on with Hashem's help. Rather it seems like Rabeinu Tam says that hot water into cold implies that the receiving body of water is larger than the incoming water, for it is common to put the smaller amount into the larger and therefore the smaller amount of hot water won't cook at all [not even the layer] since it gets mixed with the larger amount of cold water and it's heat is nullified. Though pouring cold water into hot is forbidden since the hot water is the larger amount, when the cold water enters it will be cooked.
The Rabanan taught, a person may put hot water into cold water but not cold water into hot water, these are the words of Beis Shammai - explains Rashi z'l since the lower one wins. This does not seem correct, for if this is true it should be a question on the one who says in Pesachim that the upper one wins, for the limit of Beis Hillel's argument [with Beis Shammai] is if one pours it into a cup which is a kli sheini as we say nearby, and further, even according to the one who holds the bottom one wins, he admits that [the incoming liquid] affects a layer of the bottom one, since before the bottom one can cool off the top one, the top one effects the bottom one, and if so, here too it would be tamei since the top one cooks a layer of the bottom one and assurs it. Rather this is the [correct] explanation: a person may put hot water intocold water since they cool off when they are poured and they get fully mixed into the cold ["lower" water] and they have no power to cook [at all] even if they are coming from a kli rishon. For even if you would want to say that water poured from a boiling kli rishon is as effective as a kli rishon itself, that is only true when one pours it onto spices or the like where the hot water is not absorbed into them, rather the flow of liquid hits them and cooks them, though when mixing water together which is a true mixture of the 2 items, the stream of poured water does not have the strength to cook the lower water.
(יב) מיחם שפינה ממנו מים חמין מותר ליתן לתוכו מים צונן [מרובים] כדי להפשירן ומותר לצוק מים חמין לתוך מים צונן או צונן לתוך חמין והוא שלא יהיו בכלי ראשון מפני שמחממין הרבה: הגה ואם המים מרובים כל כך שאי אפשר שיתבשלו רק שיפיגו צינתן אפי' בכלי ראשון שרי רק שלא יהיה על האש [טור]:
(12) 12. A hot water urn that has been emptied may have (a large amount of) cold water poured into it in order to warm the water. It is permissible to pour hot water into cold water or to pour cold water into hot water as long as they are not in a Keli Rishon as the water there is very hot. RAMA: If there is a large amount of water (being poured into the hot) to the extent that the water will not get cooked, instead the water will merely lose its chill, and it is permitted (to do so) even into a Keli Rishon as long as it is not on the fire. [Tur]
(פג) (פג) ואם המים מרובין - קאי אצונן לתוך חמין וכן אם חמין לתוך צונן היה הצונן מרובים שרי. ועין בח"א שכתב דבצונן לחמין דוקא ששופך בשפיכה גדולה הכל בפעם אחת אבל מעט מעט אסור לערות שהרי מבשל תיכף ומה יועיל מה שמצטנן אח"כ:
If the water is a large quantity - this is talking about the case of cold water into hot water. [If there is a lot more incoming cold water than receiving hot water everyone will agree that the cold water will win and not be cooked, so it will be permitted on Shabbos] so too, if the hot water is poured into a larger quantity of cold water it would be permitted. Look at the Chayei Adam who writes that cold water into hot water [only works if it is a case in] which you pour a large amount, or all, of the cold water at once [so it can overwhelm the hot water] though if one pours only a little bit at a time, it is forbidden, for it will cook immediately, and what will it help that later on it will cool down. [Meaning, if one pours a large amount of cold into the hot, we say that it overwhelms that hot water and prevents it from cooking the incoming cold water, but if one pours in the cold water in small amounts, even though once all the cold water mixes with the hot water it will cool it down, but when the first bit is poured, it will be cooked and a forbidden act of bishul.]
but in a bathtub etc. - it would seem that this tub is referring to when it is a kli sheini and it is not like the tub discussed earlier, where he said, I wanted to put a small container into the tub... since we said that a pitcher is not like a tub, and this pitcher must be a kli sheini, for if it was a kli rishon why wouldn't it be like a tub? and like it the tub should be a kli sheini? Furthermore, of we are talking about a tub which is a kli rishon, we should not have discussed the case of a tub, rather we should have discussed an urn/kettle since it, like the cup, is used for drinking. Though since we discussed the tub, it implies that we are talking about a kli sheini and it is coming to differentiate in between the kli sheinis; between the cup which is used for drinking, and the tub which is a much hotter water. Since it is so hot, there is room to forbid it in order to prevent one who sees it is so hot and is convinced that it is a kli rishon. In the mishnah which does say that in the case of the urn which is a kli rishon you should not put water into it, we could have said instead the case of the tub which is a kli sheini and you shouldn't put water into it, but since the mishnah was discussing drinking water it differentiated between the kettle and the cup.
Though a bathtub - which is a kli rishon
(יא) אמבטי [פי' כלי שרוחץ בו] של מרחץ שהיא מלאה מים חמין [אע"פ שהוא כלי שני] [תוס' והרא"ש וטור] אין נותנין לה מים צונן שהרי מחממין הרבה אבל נותן הוא ממים חמין שבזה האמבטי לתוך אמבטי אחר של צונן:
(11) 11. The bathtub of a bathhouse, which is full of hot water (even though it is a Keli Sheini) [Tosefos, the Rosh and the Tur] one may not add into it cold water because its waters are extremely hot. However, one may pour hot water from this tub into another tub containing cold water.
(עח) (עח) אבל נותן וכו' - ואפי' אם האמבטי הוא כ"ר מותר לערות לתוך מים צוננים אם הם מרובין על החמין ואע"ג דאסור לערות על תבלין כמ"ש בס"י דהוי ככלי ראשון זהו דוקא על דבר גוש אבל הכא שאני שמתערבין החמין עם הצונן ומתבטל חמימותן כיון שהצונן מרובים. וליתן חתיכת בשר רותח לתוך רוטב צונן אסור דכיון שאינו מתערב מבשל כדי קליפה [מ"א]:
Though one may put - even if the tub is a kli rishon it is permitted to pour [its contents] into cold water if they are more than the hot water....
(עז) (עז) אין נותנים לה מים צונן - עיין באחרונים שכתבו דדוקא מים מועטין שאפשר שיתבשלו שם אבל אם נותן הרבה כ"כ עד שהחמין נעשים פושרים עי"ז שרי וכמו לקמיה בסוף סי"ב ונראה דאם היה האמבטי כלי ראשון יש לחוש לדעת הר"ן ורבינו ירוחם שכתבו דבאמבטי אסור ליתן אפי' צונן מרובים:
One may not add cold water to it - look in the Acharonim who write that specifically a small amount of water which can get cooked there [in the tub of boiling water is forbidden] though if one puts such a large quantity of cold water in, that the bath water turns warm from it, it is permitted, similar to what is said later at the end of seif 12. It would seem that if the tub was a kli rishon one should be wary of the opinion of the Ran and Rabeinu Yerucham who write that in a tub it is forbidden to add even a lot of cold water.