It would be helpful to consider the text from Verses 11 through 16 with Rashi’s commentary, which you may do (for example) at the Chabad website.
Verse 11 says: And he shall wave the Omer before the Lord so that it will be acceptable for you; the Kohen shall wave it Mimacharas Hashabbos = on the day after the rest day. Echoing the discussion in the Talmud (Menachos 66a), Rashi remarks that the words Mimacharas Hashabbos refer not to Sunday (the day after Saturday) but to the second day of Passover; for if you say that the word Shabbos of this verse refers to Saturday, you would not know which Saturday is intended.
At Verse 15 Rashi repeats the observation that by the word Shabbos the verse actually intends the first day of Passover.
Incidentally, if you read verses 24-32 in the same chapter (Leviticus 23), you see Rosh Hashanah (the first day of the seventh month) described as Shabbos, and Yom Kippur (the tenth day of the seventh month) described as Shabbos Shabbasoin. Neither Rosh Hashanah nor Yom Kippur is usually on Saturday, so it seems pretty clear that the verses use the expression Shabbos in the sense ("holiday") that Rashi and Menachos give it at Verses 11 and 15.