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We are beginning siman 7. The Chayei Adam writes that if one does not have wine nor chamer medina for havdalah, one can rely on the havdalah performed in shul. When it comes to kiddush, we have issues regarding kiddush b’mokom seudah (shiur S0185), there is no such requirement when it comes to havdalah so one can be yotzei in shul.
Chamer medina refers to beverages which can be alternatives to wine in situations where a person wishes to show honor or respect to another person. A drink which is only drunk because a person is thirsty is not included in chamer medina. Therefore, Rav Moshe holds that drinks such as cola or orange juice are not included in chamer medina. (See more, shiurim S0162-0163).
The Chayei Adam will come back to havdalah on a kos, but first goes back to discussing the end of Shabbos in terms of tosefes Shabbos and melacha. The Chayei Adam writes that even though certain issurei derabanan are permitted bein hashmashos on Erev Shabbos (see shiurim S0124-0126), they are prohibited during bein hashmashos of Motzei Shabbos. One could argue that in the same way that, on Friday night, these issurei derabanan are permitted during bein hashmashos, because bein hashmashos is a time of safeik and we are meikil, it should apply to bein hashmashos on Motzei Shabbos as well. However, the Chayei Adam holds that all issurei derabanan are assur, even amira l’achum which is only speech.
The Chayei Adam explains that on Friday, when it is unclear whether Shabbos has begun, we do not have to be machmir out of safeik on these issurei derabanan. However, once Shabbos has been established, we do not remove the status of Shabbos during bein hashmashos until it is clear that it is no longer Shabbos, so there is no room to be lenient during bein hashmashos.
The Piskei Mishnah Berurah points out that there is an opinion which disagrees, and holds that these periods have the same halachic status. However, the Mishnah Berurah holds that practically, we follow the opinion of the Chayei Adam.
Summary
One who does not have wine nor chamer medina for havdalah should be yotzei with havdalah in shul.
Although there are certain issurei derabanan which are muttar on Friday bein hashmashos going into Shabbos, they are all assur during bein hashmashos leaving Shabbos, and remain assur until it is certain it is no longer Shabbos.