Why is Rosh Chodesh Nisan for kings - not Rosh Hashana?

Question

Why is the Rosh Hashana for kings not the same Rosh Hashana of the year when we coronate the King of Kings? See Rosh Hashana (1:1)

בְּאֶחָד בְּנִיסָן רֹאשׁ הַשָּׁנָה לַמְּלָכִים וְלָרְגָלִים. בְּאֶחָד בֶּאֱלוּל רֹאשׁ הַשָּׁנָה לְמַעְשַׂר בְּהֵמָה. רַבִּי אֶלְעָזָר וְרַבִּי שִׁמְעוֹן אוֹמְרִים, בְּאֶחָד בְּתִשְׁרֵי. בְּאֶחָד בְּתִשְׁרֵי רֹאשׁ הַשָּׁנָה לַשָּׁנִים וְלַשְּׁמִטִּין וְלַיּוֹבְלוֹת, לַנְּטִיעָה וְלַיְרָקוֹת. ‏

The four new years are: On the first of Nisan, the new year for the kings and for the festivals; On the first of Elul, the new year for the tithing of animals; Rabbi Eliezer and Rabbi Shimon say, on the first of Tishrei. On the first of Tishrei, the new year for years, for the Sabbatical years and for the Jubilee years and for the planting and for the vegetables.

Answer 1

As we learn in the first Bartenura of the first Mishna in Rosh Hashana, the Rosh Hashana for kings is on Rosh Hashana!

The exception being for Jewish kings, and the reason is that we learn from a verse (in Melachim-1:6) that we count their reign using the Exodus-calendar, which starts in Nissan.

אַרְבָּעָה רָאשֵׁי שָׁנִים הֵן. לַמְּלָכִים. מַלְכֵי יִשְׂרָאֵל מוֹנִין לָהֶם מִנִּיסָן. שֶׁאִם מָלַךְ מֶלֶךְ בִּשְׁבָט אוֹ בַּאֲדָר, מִשֶּׁהִגִּיעַ נִיסָן כָּלְתָה לוֹ שָׁנָה וּמַתְחִילִין לִמְנוֹת לוֹ שָׁנָה שְׁנִיָּה. וּמִקְּרָא נָפְקָא לָן דְּמִנִּיסָן מָנִינַן לְהוּ, דִּכְתִיב (מְלָכִים א ו) וַיְהִי בִשְׁמוֹנִים שָׁנָה וְאַרְבַּע מֵאוֹת שָׁנָה לְצֵאת בְּנֵי יִשְׂרָאֵל מֵאֶרֶץ מִצְרַיִם בַּשָּׁנָה הָרְבִיעִית בְּחֹדֶשׁ זִו הוּא הַחֹדֶשׁ הַשֵּׁנִי לִמְלֹךְ שְׁלֹמֹה עַל יִשְׂרָאֵל, מֵקִישׁ מַלְכוּת שְׁלֹמֹה לִיצִיאַת מִצְרַיִם, מַה יְּצִיאַת מִצְרַיִם מִנִּיסָן מָנִינַן לַהּ, אַף לְמַלְכוּת שְׁלֹמֹה מִנִּיסָן מָנִינַן לַהּ. וּלְמַלְכֵי אֻמּוֹת הָעוֹלָם מוֹנִין מִתִּשְׁרֵי, וְהַיְנוּ דִתְנַן לְקַמָּן בְּסָמוּךְ בְּאֶחָד בְּתִשְׁרֵי רֹאשׁ הַשָּׁנָה לַשָּׁנִים, כְּלוֹמַר לַשָּׁנִים שֶׁל מַלְכֵי אֻמּוֹת הָעוֹלָם.‏

Jewish kings: From the month we became a nation, at the Exodus.

Non-Jewish kings: From when the world was created.

Answer 2

So what is a "new year" for Kings anyway?

It was used in formal documents (Shtar) when a date was given. A Ketuvah, Get or any other document that requires a date. (Source, Gemara Avoda Zara... not sure which Daf).

So the document would say something like: On the 19th day of the 3rd month, in the 4th year in the reign of rule of King Chezekiah of Judah....

That doesn't explain though why it is Nissan:

• Where the verse refers to Solomon building the Temple, it mentions that it was the 4th year of his reign and also 480 years from the Exodus. This suggests that both of these years start on the same date
• In the verse (Kings I, 6:1), it says "...the month of Ziv, the second month". Of course we don't call any of the months Ziv but tradition is that it is Iyar (not Cheshvan). Thus the verse is telling us, by stating both the name and number, which the second month is (and thus which the first month is).

So draw the source either from the mention of the Exodus in that verse or the name of the month.