Some of these make sense in context - I prefer birthing mother, but I can see birthing person makes sense in some situations such as when the person giving birth doesn't intend to continue as the active mother afterwards
People who menstruate can make sense over just using women, particularly if your target market includes girls and/or discussing something specific to do with menstruation which won't involve all women or girls. It has been used in some spaces to avoid saying women, which helps no one but those who want to feel smug about it.
Calling anyone womb carrier or cervix haver is dehumanizing, and can be worded far better for situations where the need only impacts those with a uterus and/or cervix. I struggle to think of any situation where partner with eggs is relevant. These all come across as ways to avoid saying women.
But why not say "woman who menstruates" or "mother who gave birth" then? The people who do these things are not subsets of "people" in general, they are subsets of the specific half of people who are female, aka "women".
Girls are also female, some of them menstruate, and girls who are mothers are the group of mothers who are significantly at the highest risk across the board. I wouldn't use women who menstruate unless I was only talking about adults - the idea that menstruation or birth makes a girl a woman, makes them adults, is something I actively avoid in my language.
The only reason to use the word "people" is to promote the neo-sexist belief that womanhood is a personality trait not a physical fact with physical consequences.
Or because people is shorter than saying girls and women. The narratives in many cultures that menstruating or giving birth as something only women do, and makes girls into women regardless of their age is something I think causes significant harm, far more than saying people.
I've also used it when working with young people who are very uncomfortable with the physical facts of their bodies and meeting them where they are at. While language is important, I don't think all the fights around language need to happen all the time with all audiences - sure, with the organisations people have discussed, have at them, but when talking to a girl who has been through things many of us have and hates being seen as a girl as a consequence, neutralising the language to their comfort level can often be the first step in getting them to discuss those physical facts, consequences, and health information they need. Pushing the sex-based language at all times can create barriers and entrench them against the realities of their bodies.
While I see the treating sex as a personality trait in older adult rhetoric (usually the older men), I don't see a lot of the young people who push against discussing their sex view it as a personality trait, at least IME. It's more part of reality that has caused harm to them that they don't yet have tools to deal with and often hide away in fantasies where they can make it go away. Neutral language within appropriate context, like person who menstruates as part of discussing the issues they are facing with menstruation, can be part of developing those tools and starting to develop at least a tolerance towards that part of reality that they don't like.