What is meant by "pure" oil for Hanukka?

Question

The concept of "pure" oil is central to the holiday of Hanukka (see, for example, https://www.chabad.org/holidays/chanukah/article_cdo/aid/221817/jewish/Pure-Oil.htm and https://www.aish.com/h/c/ and https://www.ou.org/holidays/vintage-vessel/).

What makes some oil pure enough for use in the Temple to light the menorah and other oil not pure enough for this purpose? (Surely this has been asked before, but I did not find the question when I searched a few ways.)

Clarification and update: I relied on the English word "pure" when asking the question. But I am learning there are various Hebrew words for the word pure (e.g., relating to source of the oil, clarity, tahor), and the pasuk concerning lighting the Temple Menorah uses zach, a word that refers to physical purity rather than spiritual (tahor). But the quote provided by @DannySchoemann seems to refer to lack of spiritual purity (i.e., tamei), so perhaps the encompassing nature of the English word pure is a useful translation, capturing all aspects of oil that would make it not suitable for use.

In any case, I found two webpages that discuss the topic of my question:
https://www.ohr.edu/this_week/talmud_tips/7148
https://www.oxfordchabad.org/templates/blog/post.asp?aid=708481&PostID=96756&p=1

Answer 1

As noted, there are two types of purity required in the oil for the Menorah: clarity of the oil, שמן זית זך, as elaborated upon in the mishna in Menachos, and ritual purity, טהרה, that is needed for all offerings in the Temple of any sort. These two concepts together contributed to the dilemma leading to the Chanuka miracle. Since the oil needed is of a very specific grade of clarity, there wasn't very much of it sitting around so that the Maccabees could just go to any old storehouse; it was only in the Temple storage areas. Those stores had been contaminated by the Greeks with ritual impurity, and only the one sealed vessel was still ritually pure. In order to attain oil that was both of the correct type and also ritually pure, they would have to travel eight days round trip, or wait until they could purify themselves through the seven day purification process and then produce new oil (see B.Y. 680).

Answer 2

The concept "pure oil" with regards to lighting the Menora in the Mikdash is explicitly defined in Menachot Chapter 8 Mishnayot 4 & 5.

Basically, the olives are squeezed but not crushed, producing oil that doesn't need to be filtered.
This fulfills the Biblical command of שמן זית זָך - pure olive oil - in that it was always pure and didn't need filtering to become so.

Here's the original, with Sefaria's unedited translation. ("candlestick" meaning the menora in the Mikdash)

ה זֵיתִים, וּבָהֶן שְׁלשָׁה שְׁלשָׁה שְׁמָנִים. הַזַּיִת הָרִאשׁוֹן, מְגַרְגְּרוֹ בְרֹאשׁ הַזַּיִת וְכוֹתֵשׁ וְנוֹתֵן לְתוֹךְ הַסַּל. רַבִּי יְהוּדָה אוֹמֵר, סְבִיבוֹת הַסַּל. זֶה רִאשׁוֹן. טָעַן בְּקוֹרָה, רַבִּי יְהוּדָה אוֹמֵר, בַּאֲבָנִים. זֶה שֵׁנִי. חָזַר וְטָחַן וְטָעַן, זֶה שְׁלִישִׁי. הָרִאשׁוֹן לַמְּנוֹרָה, וְהַשְּׁאָר לַמְּנָחוֹת. הַזַּיִת הַשֵּׁנִי מְגַרְגְּרוֹ בְרֹאשׁ הַגַּג, וְכוֹתֵשׁ וְנוֹתֵן לְתוֹךְ הַסַּל. רַבִּי יְהוּדָה אוֹמֵר, סְבִיבוֹת הַסַּל, זֶה רִאשׁוֹן. טָעַן בְּקוֹרָה, רַבִּי יְהוּדָה אוֹמֵר, בַּאֲבָנִים, זֶה שֵׁנִי. חָזַר וְטָחַן וְטָעַן, זֶה שְׁלִישִׁי. הָרִאשׁוֹן לַמְּנוֹרָה, וְהַשְּׁאָר לַמְּנָחוֹת. הַזַּיִת הַשְּׁלִישִׁי, עוֹטְנוֹ בְתוֹךְ הַבַּיִת עַד שֶׁיִּלְקֶה, וּמַעֲלֵהוּ וּמְנַגְּבוֹ בְרֹאשׁ הַגַּג, וְכוֹתֵשׁ וְנוֹתֵן לְתוֹךְ הַסַּל. רַבִּי יְהוּדָה אוֹמֵר, סְבִיבוֹת הַסַּל, זֶה רִאשׁוֹן. טָעַן בְּקוֹרָה, רַבִּי יְהוּדָה אוֹמֵר, בַּאֲבָנִים, זֶה שֵׁנִי. חָזַר וְטָחַן וְטָעַן, זֶה שְׁלִישִׁי. הָרִאשׁוֹן לַמְּנוֹרָה, וְהַשְּׁאָר לַמְּנָחוֹת:‏

There are three [periods of gathering in the] olives and each crop gives three kinds of oil. The first crop of olives is when the olives are picked from the top of the tree; they are pounded and put into the basket. Rabbi Judah says: around the basket. This gives the first oil. They are then pressed with the beam Rabbi Judah says: with stones. This gives the second oil. They are then ground and pressed again. This gives the third oil. The first [oil] is fit for the candlestick and the others for menahot. The second crop is when the olives at roof-level are picked from the tree; they are pounded and put into the basket. Rabbi Judah says: around the basket. This gives the first oil. They are then pressed with the beam Rabbi Judah says: with stones. This gives the second oil. They are then ground and pressed again. This gives the third oil. The first [oil] is fit for the candlestick and the others for menahot. The third crop is when the last olives of the tree are packed inside the house until they become overripe; they are then taken up and dried on the roof they are pounded and put into the basket. Rabbi Judah says: around the basket. This gives the first oil. They are then pressed with the beam Rabbi Judah says: with stones. This gives the second oil. They are then ground and pressed again. This gives the third oil. The first [oil] is fit for the candlestick and the others for menahot.