The M”B in Siman 681 Sif-Katan 1 brings from the Bach and Ateres Zakanim that one may use the “Ner Channukah” first for Havdallah, put it out, then light it again for Channukah. This is even better since once you’ve done one Mitzvah with this Ner, another Mitzvah should also be done with it.
If they are speaking about the normal candle used for Havdallah which is supposed to be “avukah” (like a torch — See OC Siman 298 Sif 2) then how could this be used for Channukah afterwards? The Ner channukah shouldn’t be like a torch — or like the wording of the poskim over here in Hilchos Channukah, like a madurah. (See Siman 671 Sif 4 in the Rema.) I’m assuming whether this candle would be called avukah or a madurah for the same reason it wouldn’t be good for Ner Channukah.
Seemingly the language that the Mishneh Brurah in Hilchos Channukah Siman 681 uses that one is allowed to first light the “Ner Channukah” for Havdallah is telling us that this is a “normal Ner”, kosher to use for Channukah. However if that’s the case (and there is no difficulty with how a avukah candle is being used for Ner Channukah) why would the Bach and Ateres Zakanim want to give up the milah of avukah by the ner for Havdallah in order to get the advantage of using one ner for 2 mitzvos?
(And to say that this is the only Ner that is around to use for Channukah and Havdallah would be difficult. Otherwise why would the M”B say over that there is an advantage of using the same ner?)
I assume that the Mishnah Berurah is referring to somebody who takes two Chanukah candles and uses them together to form an avukah for havdallah.
After using them both for havdallah, there is an extra virtue of using them for the Chanukah lights, because they have already been used for a mitzvah.