That is correct, but Jewish in this context is an ethnic group, not a religion. Much of the original Zionist movement was comprised of secular Jews.

"Explicitly jewish" could be a democratic state with jewish majority. And it could be a jewish minority ruling over a non jewish majority.

Most Israel want the former. A hard core minority wants the latter. The last few decades the latter set the tone because the dovish side of the map hadn't propose a viable course of action to change the status quo. The moderate majority is too scared and it lost faith in trying to reach a peace agreement again after the violence the last try brought, And the "death to israel" rhetoric of palestinian leaders.

So while the majority is pretty moderate, the perceived lack of partner basically put in power an extreme right minority. This might change as there's an undercurrent of population change, where the new majority might be less preoccupied with western values of democracy and citizen rights.