A number of times in davening, we find the word שאתה, which I would assume would be vowelized as שֶׁאַתׇּה, "she'atah", that you. Instead, we find it vowelized as שָׁאַתָּה, "sha'atah".
For example, in Modim (siddur link):
מודִים אֲנַחְנוּ לָךְ. שָׁאַתָּה הוּא ה' אֱלהֵינוּ וֵאלהֵי אֲבותֵינוּ לְעולָם וָעֶד
I realize that the source for this must be Shoftim 6:17, which vowelizes it this way:
וַיֹּ֣אמֶר אֵלָ֔יו אִם־נָ֛א מָצָ֥אתִי חֵ֖ן בְּעֵינֶ֑יךָ וְעָשִׂ֤יתָ לִּי֙ א֔וֹת שָׁאַתָּ֖ה מְדַבֵּ֥ר עִמִּֽי׃
And Rashi there comments (Alhatorah translation):
שאתה – כמו שאתה.
"That you." The same as שֶׁאַתׇּה, with the segol vocalization.
Why would the siddur follow what is a strange spelling which needs to be explained in Tanach (sha'atah), as opposed to what would be expected (she'atah)?
Bonus: which is correct for davening?