> This probably is the central question on this topic: When does the gamete/zygote/embryo/fetus/infant/whatever go from being not-human to being human?

I disagree; that question is not relevant. Many people say it is relevant, but I disagree with that. Whether or not it is "human life" does not somehow mean it is better than everyone else.

What is relevant is: Conditions are not ideal, and if the child is a part of mother's body before they have been born then it is up to the mother to decide what to do with their body, to save themself, or if they prefer, to take a risk. Sometimes, the unborn child can be a danger to both themself and to their mother, although not always. It is best not to do abortion but conditions are not ideal so sometimes that might be better than the alternative. It is better to not even be pregnant at first if you do not want any children, if that is possible; but sometimes, unfortunately that does not work that way. So, sometimes the abortion will be a "necessary evil", possibly lesser than the other "evil" in some cases, because it is not as simple as they would have you to believe. You should perhaps consider something else first, but maybe you can't, for whatever reason; if you cannot, then abortion sooner rather than later will probably be better, for some reasons.

> If I live alone in the woods and wake one day to find an infant on my doorstep, am I obligated to care for it?

That is also not relevant for abortion, and they should not confuse it as such. If you are an orphanage then probably you should care for the child (if the orphanage is not completely full, I suppose), but otherwise you will have to decide yourself what (if anything) you will do (e.g. perhaps, you might try to figure out where the child came from and who put it there).

Wikipedia says, about "Judaism and abortion", that "The existing life takes precedence over the potential life." I have seen something similar before in a different context. I think that is reasonable. Whether or not it is "human life" is not the point (regardless of whether or not that is the point in the context it comes from; I am just taking that sentence for itself, in the general context of the discussion of abortion). That does not mean that potential life is worthless. It should mean, that sometimes is necessary to consider the risk and how to consider such a risk in such a case, because the existing life is considered "better" for this purpose, if necessary.

However, the above is only about if abortion should be moral; none of that is the same as considering whether or not abortion should be legal. I think that, being illegal causes more problems, and that it should be legal, even if not encouraged. (Unfortunately, sometimes making something legal will cause people to encourage it, so that is a possible argument against it.) The law could not consider the circumstances properly; people can consider their own body by themself, and any doctors they wish to consult; not the difficulty of dealing with government, police, etc. However, nobody should be allowed to give an abortion to someone who does not wish it.

(Making exceptions for rape, etc isn't the solution either, since then they would have to know if it is rape, which just adds delays and other legal issues.)

Apparently in some jurisdictions there is "double homicide" if you kill someone who is pregnant, even if abortion is legal. I think it makes sense, at least in some cases, such as if they were deliberately trying to kill the child (for their inheritance or for whatever other reason they may have had a motive to do so).

Rather, the laws should be made to reduce needing abortion; that will reduce abortion, because even if something is illegal, the people will do it anyways (possibly less than if it is legal, but still it will be done anyways).