My understanding of the second wave comes largely from reading intensively about the era and the books by feminists who were active then, and of course any broad-brush conclusions are just that, but to me the overriding argument was that being female should not mean that we are not entitled to have lives as full human beings.
That meant wanting the same opportunities and rights as men had, but also that the specific needs which come with having a female body and the role of the one who creates the next generation are properly taken into account.
As an aside, I believe that the first wave and the second wave both created significant change (though the first wave really wasn't the very first wave etc.), but that the waves after the second are more ripples than waves, i.e., history is unlikely to see them as equally significant to the second wave. So we are yet to see the true third wave, though sometimes I think it may be beginning now? Perhaps futile optimism.
Real feminism is happening elsewhere in the world, in South Korea, Thailand and so on, and what I have learned about it is that it is more like the second wave than whatever the current number of waves here might be. That would be because the initial situation resembles more the situation before the second wave in the West. -- Afghanistan etc. are probably not in that category but in needing their real first wave.
We need the word 'patriarchy', because there are still countries in which it is baked into everything from religions and laws to culture and tradition. And in some places it is getting stronger.