The Rambam lists five Mitzvot of Shabbat.
(א) לשבות בשביעי; (ב) שלא לעשות בו מלאכה; (ג) שלא לענוש בשבת; (ד) שלא לצאת חוץ לגבול בשבת; (ה) לקדש היום בזכירה
a. To rest on the seventh b. Not to work on it c. Not to punish on Shabbat d. Not to go beyond the boundary on Shabbat e. To sanctify the day with remembering
Why is it called “the seventh” in the first mitzvah; in the second referenced with a pronoun; in the third and fourth called “shabbat”; and in the fifth called “the day”?
Perhaps this distinguishes the primary mitzvot which establish the identity of the day from secondary mitzvot which are applicable once this identity exists. The mitzvot of resting and refraining from melacha are what makes the 7th day into Shabbat. Whereas the other prohibitions are responses to it being Shabbat, a day of rest. Even though God gave Shabbat Kedusha in Bereshit, that is only a potential. Its identity is actualized through the Jewish people.
Perhaps this is the meaning of the Jewish people being the “partner” of Shabbat (Ramban Bereshit 2:3 quoting Midrash Rabba). The six days match up between the creating of an environment (days 1-3, Shamayim and Aretz) and realizing it through its inhabitants (days 4-6, Kol tzevaam). For example, birds and fish of day 5, in the sea and sky of day 2. The Jewish people have this relationship in relation to shabbat.
This leaves the question of Kiddush. Why is Kiddush characterized as “the day” instead of “the day of Shabbat”?
Perhaps it points to Kiddush as relating to it as a day of rest, rather than its specific identity. Kiddush is a meta-principle of articulating the kedusha of a day as part of distinguishing it. One does not leave the kedusha to be expressed incidentally through action, rather one must explicitly characterize that time as distinct. Kiddush is not a halacha in the specific identity ofShabbat, rather of Shabbat as the paradigm of “Shabtot Hashem”, including Yom Tov. Unlike the Mitzvot of resting and refraining from work which have distinct Mitzvot for each day, Yom Tov does not have a separate mitzvah of Kiddush.