The mitzva of counting the Omer begins "mimacharat hashabbat" (Vayikra 23:11) - on the day after the "shabbat". "Shabbat" here is traditionally understood by Jews to mean the first day of Pesach, not the 7th day of the week. The question of whether this traditional interpretation is correct has been discussed to death. I have just one thought to add on the matter.
I was reading a book on Ethiopian Jewry, which said that according to their tradition, "mimacharat hashabbat" refers to the seventh day of Pesach.
Now, we know that Ethiopian "halacha" as it has reached us bears no apparent relation to the halachic traditions of the Mishna, Gemara, and later mainstream Jewish authorities. Rather, it seems to be based on a "literal" understanding of the written Torah, similar to the Karaite approach. Indeed, as we see here, the Ethiopians disagreed with the rabbinic interpretation that "shabbat" means the first day of Pesach. And yet, they did not concur with the Karaite opinion either. Independently of the rabbis, Ethiopian Jews came to the conclusion that "shabbat" actually refers to a holiday.
How did the Ethiopians reach this conclusion? I don't know. By reading the articles I linked to above, you will find a variety of arguments for this view, which is our view. I presume that the Karaites can produce a similar array of arguments in the opposite direction. Deciding the correct meaning of the verse may be the kind of judgment call which depends as much on your prior expectations and modes of thought as on the objective strength of the arguments. And I don't know what were the prior expectations of the Ethiopians, nor of the Karaites.
But let it not be said that the only tenable interpretation is that "shabbat" means Saturday, that the rabbis distorted or overrode the verse's plain meaning. We have a data point which proves otherwise. The Ethiopians had no "rabbinic agenda" and nevertheless decided that the basic rabbinic view was most reasonable.
Perhaps, as hard as it is for Karaites and certain scholars to admit this, the rabbinic tradition is not a series of wanton distortions for partisan purposes, but rather an honest attempt to integrate textual analysis and intellectual reflection with the received body of tradition.