Knighters Revenge, I think you got the generations wrong, would be more Baby Boomers for second wave feminism, but GenX would likely be in the same grouping.
Second wave feminism was, and remains, somewhat more complicated. What you describe above would be defined as liberal feminism. There were also socialist feminists and radical feminists back in the day, now the socialist feminists are a much smaller proportion but plenty of radical feminists still around, many of my friends are what would generally be referred to as radical feminists.
The whole Trans issue is very divisive and there are many opinions on this. Certainly there is a lot of criticism of trans arguments by radical feminists. Radical feminists have always viewed gender as a social construct, remembering that gender is the role you have in society, sex is biological. Certainly one of the criticisms of the trans movement is that, while they are undermining gender roles to some extent by claiming a right to identify with a gender role not linked to their biological sex, they are also conservative in the sense that they are not challenging the whole edifice of gender roles. Why should men be masculine and women feminine? Why don't we have more of a variety of gender roles so people can be comfortable with their biological sex while rejecting the traditional gender role assigned to that sex?
Intersectionality is, to some extent, a response to how the feminist movement was often seen as dominated by white middle class women. Yes, it is good to promote women to high positions, but what about the really hard yards of the millions of women who are in low pay, low status jobs which are dominated by women? Intersectionality is also not a new concept, it is a nice term in the sense that it describes a lot of situations which used to need specific detail, but the reality is that it has long been recognised that, while middle class white women are discriminated against for being women, there are a lot of women out there who are discriminated against for being women, but also face discrimination/disadvantage on the basis of race, class, caste etc.
This stuff is not simple.
However, what is simple is that no means no.
DS