What is the underlying argument between the two pronunciations and what are the multiple ways to deal with the problem?
Per the Mishna Berura "Know that some say that the word זכר should be pronounced as "Zaicher" with a "Tzairai" and some say that it should be pronounced "Zecher" with a "Segol." Therefore, it is correct to read it twice.
The Mishnah Berurah does not cite a source for the two opinions, however some have suggested that the source is the following letter by Rabbi Chaim Volozhin. Rabbi Chaim was a leading student the Gaon of Vilna, and the following is a section of a letter that was written by Rabbi Chaim to the publisher of Ma'aseh Rav, a collection of testimonies about the Vilna Gaon's personal customs.
The different ways I have seen are either totally repeating the Posuk and saying the word differently or just repeating from Timche with the other pronunciation.
See Igros Moshe for further discussion. http://hebrewbooks.org/pdfpager.aspx?req=922&st=&pgnum=111
A comprehensive article about this was written by the late Rav Mordechai Breuer, published in Megadim 10, Shvat 5750.
He discusses the "Zeicher" issue, as well as "V'laharog" and "Lifneihem" in the Megillah (discussed in Rav Moshe's teshuvah).
He traces the history of the variant pronunciation back to the Radak, and discusses the mesorah for the word from sources like the Keter and the Minchat Shai.
His conclusion is that from a masoretic perspective, "Zeicher" with a tzeireh is correct--but it's really worth reading the whole article.