Klein Dictionary gives the etymology of שַׁבָּת as derived from שׁבת (to rest). I also found references suggesting the origin is with the Akkadian šapattum, the 15th of the month, but none related to שׁבע (which originates in Proto-Canaanite).
היּוֹם הַשְּׁבִיעִי (note the definite form!) is how Saturday is referred to in Genesis 2:2-3, without actually having the name שַׁבָּת. There's a some discussion about why שַׁבָּת isn't mentioned there as a name for the day, but regardless, your comment is the first time I read of the etymology of שַׁבָּת deriving from שׁבע.
> Other related words are ... to sheva (שֶׁבַע) meaning seven, as Shabbat is the seventh day of the week ...
¹ “Fair” as in “a gathering of stalls and amusements for public entertainment”.
I can confirm that in Hebrew, the name for Sunday translates into "First Day", Monday into "Second Day", etc. Except for Saturday, which is Shabat.