Anyway, I wondered if the term "gender critical" is something some feminists have chosen for themselves, or is it a label imposed on them? And can one be gender critical without being a feminist, or vice versa?
Good question, Callixte. The term gender critical to describe those who disagree with the doctrine of gender ideology and its associated politics gained widespread popularity in response to a slur applied mostly to women who rejected the doctrine.
There's a widespread myth the slur (the abbreviation for trans exclusive radical feminist is banned here, so I'll just say "the slur") was coined by a radical feminist, but it was actually coined by a transgender person and then picked up in feminist circles whose members accepted the doctrine and wanted to distinguish themselves from feminists who didn't.
So yes, one can be gender critical without being a feminist and a feminist without being gender critical. (The most prominent strand of feminism that accepts the doctrine of gender identity is liberal feminism.)
Almost from the beginning it was used in a derogatory way, and in usage today its regularly combined with threats of violence, so those who wanted to discuss why sex and not gender should form the basis of legislation called themselves gender critical to more accurately reflect their position. As did those who reject the doctrine of gender identity not because they are working from a radical feminist perspective but from a free speech perspective - who may or may not hold feminist views.
Also, the term "radical feminist" seems to be used interchangeably with "gender critical feminist" but much of the feminist analysis I've seen that's labelled as "gender critical" from the UK seems based in Marxist, materialist or socialist feminism rather than radical feminism. Is there a difference between radical feminst versus Marxist feminist views on "gender" and the conflict of rights?
That depends entirely on the writer. Traditionally Marxist feminism shares with radical feminism the class-based analysis of the issue of women's inequality, but disagrees about the root cause (economic class vs sex).
However, there's less daylight between those two groups today than there was in the past, because both also now consider the intersection of different axes of discrimination experienced by female people (aka intersectionality, this can include sex, class, race, religion, disability, age and sexuality).
Many people now labelled radical feminist because of their opposition to the doctrine of gender identity are neither radical nor feminists but simply women's rights proponents. Women have needs, there's a law that caters to those needs, and they wish to uphold this without necessarily engaging in theoretical feminist discourse. That's where I started from.