Since the first blessing of the Shemoneh Esrei relates the praises of Hashem, these three expressions עוזר ומושיע ומגן follow an ascending order of praise of the way Hashem saves us.
There are several explanations what these terms mean, but I will bring just one of them from the sefer אבני אליהו as brought by the sefer רנת יצחק here to illustrate this order:
There are three ways to show one's trust in Hashem when one needs help, and correspondingly there are three ways in which Hashem saves him:
1) To do something (make some effort), in which case Hashem is עוזר - helps him by blessing his action with success.
2) To do nothing, in which case Hashem is מושיע - saves him without the need for the person to do anything towards his salvation.
3) To do something which goes against what is intelligent and for the sake of Hashem put himself in danger, in which case Hashem is מגן - shields all who trust in Him.
(And since Avraham trusted in Hashem to the utmost and put himself in danger many times for the sake of Hashem's honor, Hashem said to him: I will be a shield for you, and thus we conclude the first blessing "Blessed are You, Hashem, the shield of Avraham". Similarly, since King David also trusted in Hashem to the utmost, we conclude the third blessing after the reading of the Haftorah "Blessed are You, Hashem, the shield of David").
But the prayer that the gabbai recites is different because, as the sefer Rokeach and the Machzor Vitri explain, it is a continuation of the אב הרחמן prayer which contains a further six expressions of Hashem's salvation: הוא ירחם...ויזכור...ויציל...ויגער...ויחן...וימלא, making a total of nine expressions of redemption corresponding to our nine exiles. Thus, the order of the words presumably corresponds to the particular aspect of redemption that was needed for each exile.
I don't know which list of exiles they are referring to, but there is however a well known story about Reb Chaim from Volozhin in which he relates that there were a total of ten exiles - Babylon, North Africa, Egypt, Italy, Spain, France, Germany, Poland, Lithuania, and finally America. (Artscroll Biography of Reb Chaim of Volozhin, p.208)
Finally, the order of the last three expressions - the ones mentioned in the question - are the same as the order in the blessing between Krias Shema and the Shemonah Esrei of Shacharis, which talks about Hashem's saving us from Egypt, the archetypical exile:
עזרת אבותינו אתה הוא מעולם, מגו ומושיע לבניהם אחריהם.