Atah Chonantanu 3 (Klal 8 Siman 5-6) Hilchos Shabbos - S0223

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We are beginning siman 5. Chazal enacted an issur to eat--not even to taste--before reciting havdalah. We have learned that there are cases in which Chazal allow a person to taste before performing a mitzvah, but regarding havdalah, any amount of food is assur to consume before reciting havdalah on a kos. The only exception is that it is permitted to drink water. 
The issur to eat before havdalah is so serious that there is an opinion in the Gemara that if one forgot and ate before making havdalah, they are “fined” in that they can no longer recite havdalah. Although we do not pasken in accordance with that opinion, there is a situation in which it is a factor.

We learned that if a person forgot to recite atah chonantanu in shemoneh esrei, generally they do not need to repeat shemoneh esrei because they will make havdalah on a kos. If that person subsequently ate before havdalah, we learned above that, generally, they would still make havdalah. However, in our case, since they ate before havdalah and forgot to recite atah chonantanu, the fine will be applied in that they cannot rely on the havdalah on the kos to replace the havdalah in shemoneh esrei, and they must repeat shemoneh esrei. After repeating shemoneh esrei, one would then make havdalah on a kos. 

We see that even though we pasken that eating before havdalah does not prohibit someone from making havdalah on a kos, that person has still degraded, to some level, havdalah on a kos. That downgrading precludes the havdalah on the kos from being significant enough to cover the havdalah in tefillah, and therefore one must repeat shemoneh esrei.

In siman 6, the Chayei Adam writes that even if one makes havdalah on a kos before davening maariv, they still need to recite atah chonantanu in shemoneh esrei. Chazal instituted both a chiyuv in tefillah and through a kos, so both are necessary.

Summary
It is assur to eat before havdalah on a kos.
Nevertheless, if one ate before havdalah on a kos, they still make havdalah.
If, in addition to eating before havdalah on a kos, they forgot to recite atah chonantanu, the havdalah on a kos will not cover the havdalah in shemoneh esrei, and one must repeat shemoneh esrei.
Even if one makes havdalah on a kos before davening maariv, they still must recite atah chonantanu in shemoneh esrei, because Chazal instituted both types of havdalah, so both are necessary.