Another part of that belief is that a pregnant woman's body is not in fact her own, but also belongs to the baby during the pregnancy.
I think here it's where that line of reasoning takes a wrong turn, so to speak.
It is not about who owns what. When pro-lifers are against abortion, they don't say a woman's body is not her own. Instead they say the baby is its own entity even though it's connected to the mother, and even though the mother's body is her own body, she shouldn't have the right to violate the baby's body.
It's not because the baby now somehow owns the mother or that the mother owns the baby. It's all about the baby being its own entity. The mother has the rights to her body, but the baby's body is not her body, so things concerning the baby's body is not anymore about the woman's body and should be treated with different approach.
And their connection to each other - the baby being at its most helpless state without the mother as no-one else can help the baby to survive at that point - makes pro-lifers even more puzzled on how anyone can want to end that life.
You are very much right in saying that this is an issue where people have so fundamentally different beliefs that they just can't be surmounted. Discussions about this get so easily hung on words and what the other side thinks the other side means with those words.
Sometimes the discussion ends when the words zygote and fetus and baby begin to get thrown in. Pro-lifers think pro-choicers obviously should understand why a baby is so important, and pro-choicers think pro-lifers should obviously know why a zygote or fetus doesn't matter so much that they should be given the same rights as post-birth humans have.
But the reality is that the arguments from both sides still hold if you switch the words they use. Pro-lifer doesn't think any less of the new human if they have to say it's a zygote, and pro-choicer doesn't think any less of abortion rights if he has to use the word baby instead of zygote.
And then the arguments just go round and round endlessly. That's because most of the arguments from both sides don't actually address the actual issues at all. The arguments just get lost in between two very different mindsets with very little will to understand each other.