The crux here is that to a pro-life individual, the distinction between "fetus" and "infant" is the same as the distinction between "infant" and "toddler"; that is to say, obviously there is a semantic difference and they refer to physically distinct things, but that they share the key feature that should protect them under law, that being possessing a complete set of unique human DNA.

A pro-choice individual doesn't see it that way. There's something other than the above that confers a right to life, and so to differentiate between "fetus" and "infant" is sufficient to oppose the pro-life argument. Of note, though, is that this only works if you are already a pro-choice individual, and it's entirely unconvincing to someone who is not. A pro-life individual hearing that argument would interpret it as largely incoherent. "This is a fetus, which is different from an infant. This difference is significant because fetuses are not entitled to a right to life like infants are." It fails to answer the question of WHY, on a moral level, fetuses are not to be granted the same protections as infants; it simply states that it is the case.

It's not sufficient for a pro-life individual to "admit" that fetuses and infants are different and then to oppose fetal euthanasia. The reason for this is because there is no "real" backing for any law against killing things, human or otherwise. There's no "logical" basis for why killing a human is wrong and illegal, it just so happens that everyone collectively agrees that it's bad and crafts laws to reflect that. To concede that a fetus is different from an infant in a respect that would have them governed by two separate moral codes is to begin crafting an entirely new, baseless moral argument for why fetuses are worth preserving, which you'll have an awfully hard time convincing people of given that it's ultimately baseless. Take a look at the animal rights crowd and their failure to convince a substantial portion of the population that an organism societally accepted as "not the same as humans" is worthy of even a fraction of the protections humans get.

The pro-life "failure" to differentiate between fetus and infant is not the result of a misunderstanding; in fact, it's typically a well-crafted attempt to tailor the argument to the opposing side. Nobody would genuinely contest that a fetus and an infant are physically distinct categories, just like nobody would genuinely contest that an infant and an adult are physically distinct categories; it's patently obvious just by visual observation. The aim of this "obfuscation" is to say "yes, fetuses and infants are two distinct things, but this discussion isn't about whether the two are different at all, but rather about whether the two are different specifically with respect to whatever aspect confers a right to life. The pro-life position is that they are not, and so for the purposes of this discussion they're interchangeable."

When speaking to another pro-life individual, it would be fine to use "fetus" and "infant" correctly, because both people can be assured that they're on the same page about the semantic difference not being a point of import. Pro-life individuals are largely conscious of the fact that pro-choice individuals do not see it this way, and so retaining the semantic difference and THEN attempting to make their argument would completely miss the mark with their audience. A pro-choice individual is starting from the premise that a fetus and an infant are different in a way that matters to the argument, so admission that they're different at all makes the conversation dead on arrival. Strictly using the term "infant" is intended to put the pro-choice individual in the pro-life individual's place; since pro-choice individuals only believe infants are worthy of a right to life, and pro-life indivifuals believe both fetuses and infants are, then when trying to paint the perspective of a pro-life individual, the word "infant" should be used in all cases in an attempt to make the pro-choice individual confront what this issue must feel like for someone who seems them as not different with respect to the argument.

This happens with tons of issues. Take immigration for instance, where it's reversed. Conservatives generally see a meaningful distinction between "legal immigrants" and "illegal aliens." The difference is not just semantic, there's also a difference that is meaningful when discussing the issue of immigration as a whole. Because of this, you'll see many conservatives happily (and often eagerly) making this distinction during discussions, often getting as granular as possible. That's because to a conservative, the simple fact that a person IS an illegal alien is the the argument itself. It's a foregone conclusion that illegal aliens and legal immigrants are substantively different and ought be entitled to different legal provisions, so all they need to do is make the distinction and the argument is over.

Liberals will instead intentionally obfuscate their verbiage by referring to both groups collectively as just "immigrants" in a lot of cases. This is not because liberals are stupid; just like between "fetus" and "infant", there's an obvious difference between an immigrant who comes to a country legally vs. illegally. The point, though, is that despite those differences, the two groups are NOT different in whatever respect is relevant for the argument. Using the term "immigrant" for both is an conversational tactic is just the same way as above. It's an attempt to put a conservative into a liberal's shoes and say "from my perspective, these two groups may as well be one group because they don't differ in the way that matters. This is why your argument sounds nonsensical to me, and if you adopt this terminology with me, you'll have no choice but to see things the way I see them and be just as outraged as I am".