What you're talking about here is not a difference between philosophy and culture; it is about an artificial and arbitrary division of feminism into two groups by a right wing libertarian woman - Christina Hoff Sommers in a book she wrote over 20 years ago titled "Who Stole Feminism?" Supposedly, according to her treatment of history, the original feminists - the suffragettes were "equity" feminists, who just wanted the right to vote and equal treatment under the law, while the 2nd wave of feminists that rose up in the 60's were "gender" feminists, who saw the problems that women face as systemic aspects of modern patriarchal cultures, which degrade the lives of both women and men alike.
In reality, no such divide ever existed; since many of the early feminists were outright anarchists who planted bombs and wanted to bring down the whole political and economic establishment. While the kickoff for the 2nd wave - Betty Friedan, campaigned mostly around the right of middle class and higher women to be out in the workforce.
But, the myth of equity feminism and gender feminism creates a convenient trope for rightwingers of all stripes (conservative, libertarian) to attack legislation and social programs that disproportionately affect the fate of women with children, while calling themselves feminists...all because they have the rhetoric of a woman...a well compensated academic - Hoff Sommers, on their side.
The propaganda effects have been similar to what has happened to the legacy of Martin Luther King and the civil rights movement, where Republicans and rightwingers like Glenn Beck, claim to be the heirs of MLK. Part of their success no doubt, is because all sides...especially establishment liberals...started creating the myth of the moderate, unchallenging MLK as soon as he was dead and gone! So, every year to this day, American politicians and pundits can gather round the campfire and commemorate the "I Have A Dream" speech, with its selected, non-threatening passages; while never acknowledging MLK's last major speech (and likely the one that got him assassinated)
"Beyond Vietnam." What would the real Martin Luther King have to say about Barack Obama, and the political, religious and financial leaders who run the world today? I doubt they would want to hear it today any more than they did back in 1968!