What is the difference between late-term abortion and infanticide?

EDIT: When I initially posted this, I did not realize that the phrase "late-term" had a specific medical meaning that is not relevant here. I should have phrased this question: "What is the difference between an abortion on a viable fetus and infanticide?"

I know that there is an argumentative technique where you pretend that you don't understand your opponent's point of view and ask them to explain it, but that's not what I'm doing here. I genuinely don't understand this.

There are many pro-choicers who believe in abortion only until the fetus is viable. I understand them. I may not agree with them, but I totally understand their reasoning.

What I don't understand is people who believe that abortion should be legal after the fetus can survive outside the womb. I mean, an abortion starts with a pregnant woman and an abortion doctor, and ends with a non-pregnant woman, an abortion doctor, and a dead fetus. There are two ways to get from the start to the finish: Either kill the fetus and then remove it, or remove the fetus and then kill it. The end result is exactly the same. Why should it matter what order the steps take place in?

I've asked this question before, and the two answers I've gotten are:

1. "Because one is an abortion and the other isn't." But this doesn't answer the question, it just defines the terms.
2. "Because pro-lifers would lose their shit if we did it the second way." Well, yes, but that's pro-lifers. I want to know why you feel it should always be done the first way.

Obviously, removing the fetus alive and then killing it is illegal in (I believe) every country in the world. But, if some part of the world made it legal to perform abortions that way, would you be in favor of that or against it? And if you're against it, why? Explain exactly how it's different from an abortion on a viable fetus.

Please try to avoid getting off-topic. The purpose of this thread is not to discuss abortion in general, or the consequences of rape, or any of that. All I'm looking for is an answer to the question above. Thank you.

(Note: I have only a limited amount of time to be on the internet, so if I disappear for a couple of days, that's normal for me.)

EDIT 2 and 3: I would also like to add the stipulation that the fetus is healthy. There are third-trimester abortions that are performed on fetuses which are dying or will die shortly after birth, but those are outside the scope of what I intended for this discussion, and, as one person pointed out, at that point an abortion would (or at least could) be considered palliative care.

EDIT 4: And the mother's life is not at risk, either.