Rosh Hashana 7a - First Month of the Year

Nissan is considered to be the Rosh Hashana l'chadashim, meaning it is the first month of the year. The Ritva is bothered, according to R' Eliezer that the world was created in Tishrei, the count of the months began in Tishrei, shouldn't Tishrei be the first month? The Ritvah explains that until Yetzias Mitzrayim, Tishrei was considered to be the first month. Only after the exodus "gazar hakadosh baruch hu limnos m'nissan zeicher l'yitzias mitzrayim". Therefore, when the Torah was given (post yetzias mitzrayim), it considers the first month to be Nissan rather than Tishrei. This was introduced in mitzrayim when Hashem said "Hachodesh hazeh lachem rosh chadashim, rishon hu lachem l'chadshei hashana", meaning for you it is the first of months, but Adam Harishon counted from Tishrei. The same idea is apparent from the Ramban in Bo, but the Ramban is not as clear in describing that prior to yetzias mitzrayim we counted from Tishrei. The Rashbam in Bo also expresses the same idea, that Nissan is the first month for the Jews, but the rest of the world counts from Tishrei (or at least should count from Tishrei). This is the reason we count for Jewish kings from Nissan, but non-jewish kings from Tishrei (3a). The Ran (2a) elaborates that without yetzias mitzrayim there would be no possibility of a Jewish kingdom, therefore we connect the malchus of Shlomo to yetzias mitzrayim, since yetzias mitzrayim introduced the possibility of a Jewish king, so all Jewish kings commemorate this by renewing their year in Nissan.