Based on the above, it is possible to explain the connection between the two interpretations of the name Yom HaKippurim:וְעַל פִּי זֶה יֵשׁ לְבָאֵר הַקֶּשֶׁר דִּשְׁנֵי הַפֵּרוּשִׁים בְּיוֹם הַכִּפֻּרִים –
a) that on this day, atonement is granted for all undesirable conductשֶׁבְּיוֹם זֶה מִתְכַּפְּרִים כָּל הָעִנְיָנִים הַבִּלְתִּי רְצוּיִים
(the simple meaning of the word kippurim, “atonement”),(כִּפֻּרִים כִּפְשׁוּטוֹ),
and b) that this day resembles Purim.וְשֶׁיּוֹם זֶה הוּא כְּמוֹ פּוּרִים
(The resemblance between Yom Kippur and Purim is(שֶׁהַדִּמְיוֹן דְּיוֹם הַכִּפֻּרִים לְפוּרִים הוּא
that on Yom Kippur lots were cast.שֶׁבְּיוֹם הַכִּפֻּרִים הִפִּילוּ גּוֹרָל,
This resembles Purim because the name of the holiday recalls the pur, Haman’s casting of lots.)כְּמוֹ פּוּרִים עַל שֵׁם הַפּוּר הוּא הַגּוֹרָל) –
For complete atonement for conduct that runs contrary to G‑d’s will as manifest in the Torah and its mitzvos is attainedכִּי שְׁלֵמוּת הַכַּפָּרָה עַל הָעִנְיָנִים שֶׁעָשׂוּ הֵפֶךְ הָרָצוֹן דְּתוֹרָה וּמִצְוֹת הוּא
when atonement is drawn down from G‑d’s infinite light, the “Master of the will”כְּשֶׁהַמְשָׁכַת הַכַּפָּרָה הִיא מֵאוֹר אֵין סוֹף בַּעַל הָרָצוֹן
(a level that transcends G‑d’s will as manifest in the Torah and its mitzvos).1(שֶׁלְּמַעְלָה מֵהָרָצוֹן דְּתוֹרָה וּמִצְוֹת).
From the revealed levels of G‑d’s will, the lines are defined. A mitzvah is a mitzvah and sin is sin, a violation of His will. However, as Chassidus explains,2 “The source for the potential for teshuvah is because the Essence of Or Ein Sof is not, Heaven forbid, limited in His will. Although the desire for the mitzvos stems from the inner dimension and the essence of Or Ein Sof, He is not limited or compelled by [His] will, Heaven forbid.”
The intent is that as G‑d’s will is expressed, there is a distinction between good and evil, what He desires and what He does not desire. This distinction is, however, entirely dependent on His will. He, however, is the Master of His will and can grant forgiveness and atonement even when His will has been violated.
This is the connection between the two interpretations of the name Yom Kippurim.וְזֶהוּ הַקֶּשֶׁר דִּשְׁנֵי הַפֵּרוּשִׁים בְּיוֹם כִּפֻּרִים,
The fact that Yom Kippur resembles Purim whose name recalls the pur, the lottery,דְּזֶה שֶׁיּוֹם הַכִּפֻּרִים הוּא כְּמוֹ פּוּרִים עַל שֵׁם הַפּוּר הוּא הַגּוֹרָל,
indicates that on Yom Kippur as well Divine energy is drawn down from the level of “the Master of the will”הַיְנוּ שֶׁגַּם בְּיוֹם הַכִּפֻּרִים הוּא הַהַמְשָׁכָה מִבַּעַל הָרָצוֹן
(as it is drawn down on Purim).(דֻּגְמַת הַהַמְשָׁכָה דְּפוּרִים),
This serves as a rationale and an explanation why Yom Kippur is “the Day of Atonement,”הוּא טַעַם וּבֵאוּר עַל זֶה שֶׁהוּא יוֹם כִּפֻּרִים כִּפְשׁוּטוֹ,
the day on which atonement is granted for undesirable conduct.שֶׁבּוֹ מִתְכַּפְּרִים הָעִנְיָנִים הַבִּלְתִּי רְצוּיִים.
On the basis of the above, it can be understood that the atonement granted on Purim is loftier than the atonement granted on Yom Kippur.3וְזֶהוּ מַה שֶּׁהַכַּפָּרָה דְּפוּרִים הִיא נַעֲלֵית יוֹתֵר מֵהַכַּפָּרָה דְּיוֹם הַכִּפֻּרִים,
For atonement is granted on Yom Kippur only when a person repents,4דְּיוֹם הַכִּפֻּרִים אֵינוֹ מְכַפֵּר אֶלָּא עִם הַתְּשׁוּבָה
while the atonement that was granted to all the Jewish people during the days of Purim is associated with a day of rejoicing.5וְהַכַּפָּרָה שֶׁהָיְתָה לְכָל הַיְּהוּדִים בִּימֵי הַפּוּרִים הֲרֵי זֶה יוֹם שִׂמְחָה,
Purim possesses such an advantage because fundamentally, the concept of a lotteryכִּי עִקַּר עִנְיַן הַגּוֹרָל
(that energy is drawn down from the level of “the Master of the will”),(הַמְשָׁכַת בְּחִינַת בַּעַל הָרָצוֹן)
is intrinsic to Purim.הוּא בִּימֵי הַפּוּרִים,
These days are given their name because of the lottery.דְּהַיָּמִים עַצְמָם נִקְרָאִים בְּשֵׁם פּוּרִים עַל שֵׁם הַפּוּר,
Therefore, the atonement granted on Purim has no limitations.6וְלָכֵן, הַכַּפָּרָה שֶׁהָיְתָה בִּימֵי הַפּוּרִים אֵין בָּהּ הַגְבָּלוֹת,
Yom Kippur is only “like Purim,”7וְיוֹם הַכִּפֻּרִים הוּא כְּפוּרִים
merely, a resemblance.בְּכָ"ף הַדִּמְיוֹן בִּלְבָד,
Thus, on Yom Kippur, atonement is granted only when accompanied by teshuvah.וְאֵין יוֹם הַכִּפֻּרִים מְכַפֵּר אֶלָּא עִם הַתְּשׁוּבָה.

Summary

Based on the above, it is possible to explain the connection between the two interpretations of the name Yom HaKippurim: a) that on this day, atonement is granted for all undesirable conduct (the simple meaning of the word kippurim, “atonement”), and b) that this day resembles Purim, transcending the structure of Seder HaHishtalshelus, as reflected by the fact that both days are associated with the casting of lots. Complete atonement for conduct that runs contrary to G‑d’s will is attained when atonement is drawn down from G‑d’s infinite light, the “Master of the will” (a level that transcends G‑d’s will), similar to a lottery that trascends will. The atonement that was granted to all the Jewish people during the days of Purim was superior to that of Yom Kippur because the concept of a lottery is intrinsic to the day.