Which is quite much exactly what I said.
That a definition is arbitrary doesn't mean that it is unimportant. If I use one arbitrary definition, and you another one, we won't have any meaningful discussion. Sometimes this can be resolved, when the definitions themselves aren't the central issue and we can agree about at least some provisional defintions. Sometimes the definition itself is the central issue, and can't be resolved. Like when we disagree about the meaning of the term life. In this case, any provisional definitions won't help.
> then they aren't really disagreeing over the meaning of the word 'life', they're disagreeing over the nature or essence of life itself
Those two things are the the same. What is the meaning of a word other than the "nature or essense" of the thing it references?
This is actually a vexed question in linguistics and the philosophy of language. I don’t want to get into it here, but the answer to your question is not in any way uncontroversial or obvious.
For me the idea that the root of the abortion debate is a disagreement over an arbitrary definition doesn’t compute, for the reasons I already gave. (Why not just introduce new words with commonly agreed definitions and be done with it?) But that discussion seems to be going round and round in circles.
life_prolife = life begins at the conception.
life_prochoice = life begins at week 20 of the pregnancy (as an overly simplified example for the sake of argument).
Pro life: it is wrong to kill anything that has life_prolife.
Pro choice: no, it is only wrong to kill anything that has life_prochoice. It is, at least under some circumstances, ok to kill something that has life_prolife but not life_prochoice.
So of course we haven't resolved anything (which I think is your point). Rather, we have moved the discussion to which definition is the one that is relevant to use when we discuss the sanctity of life. And that is a discussion about semantics.
So the author does seem to be saying that the whole abortion debate is purely a debate over arbitrary definitions, not merely that it's important to clear up semantic issues before getting into the meat of the debate.
If the issue were purely semantic, it would be possible to resolve the debate simply by introducing unambigious terminology agreed on by both sides (e.g. using 'alive*' and 'alive†' rather than 'alive'). If this doesn't satisfy both sides, then they aren't really disagreeing over the meaning of the word 'life', they're disagreeing over the nature or essence of life itself. Informally we can say that they disagree over what it means to be alive. But philosophically it's important to be clear that this isn't just a semantic issue about the meaning of a particular word.
Another way to bring this out is to think cross-linguistically. Do we really have to have the abortion debate over again for every one of the world's ~6500 languages and their corresponding words for 'life'? I think not, because at heart the debate isn't a debate over word meaning.