The Yerushalmi in the last post starts out with a general statement which I brought, the order of proper progression of requests. It is then followed by a case by case explanation of why one is before another. The Bavli in Megilah 17b, however, interpolates the quotes with the order:
And why did they see fit to say the Ingathering of the Exiles after the blessing of the years (regarding making a living)? For it is written: "And you, mountains of Israel, your branches will give forth and your fruits you will hold for my nation Israel, for they will be coming soon" (Yehezkel 36:8).
And because the exiles have been gathered, judgement is done on the wicked as it is said: "And I will turn my hand upon you, and I will refine your dross as with lye" (Yeshayahu 1:25), and it is written (in the next verse) "And I will return your judges as at first" (Yeshayahu 1:26)
And because judgement was done on the wicked, sinners are finished, and (we) include with them intentional sinners, as it says "And the breaking of transgressors and sinners is together" (Yeshayahu 1:28)
And because sinners are finished, the horn of the righteous is uplifted, as it says "And all the horns of the wicked I will cut off, and the horns of the righteous I will raise" (Tehillim 75:11) and (we) include righteous converts with the righteous, as it says "In front of age, rise up, and glorify the face of the elderly" (Vayikra 19:32) and adjacted to that is "For a stranger will live with you" (Vayikra 19:33).
And where is this horn raised up? Their horn is in Yerushalayim, as it says "Pray for the peace of Yerushalayim, those who love her will prosper" (Tehillim 122:6)
And because Yerushalayim is built, Dovid comes (i.e. Moshiach ben Dovid) as it says (from here until the end, we continue on page 18a) "And afterwards, B'nei Yisrael will return and request Hashem their G-d and Dovid their king" (Hoshea 3:5)
And because Dovid has come, prayer will come, as it is said "And I will bring them to My Holy Mountain and I will make them happy with My House of Prayer..." (Yeshayahu 56:7)
And because prayer comes, the Service (Avodah) comes, as it says "Their elevation offerings and their sacrifices are for desire on My altar..." (Yeshayahu 56:7)
And because the Service comes, Thanksgiving comes as well, as it is said: "He who offers a thanksgiving offering honors Me" (Tehillim 50:23)
From here we see there is agreement between the Bavli and Yerushalmi on this matter that the order of the Shmoneh Esreh is the same as the order of the Geulah. The only difference of note is that the building of Yerushalayim is included with the coming of Moshiach according to the Yerushalmi.