What is your personal opinion on the matter? From looking through the site, no clear answer emerges regarding defining the fetus as a person.

My personal opinion is that I do not know from when it is a person, and precisely because of that there is a moral prohibition against aborting it at any stage. From when would there be a death penalty for murder? There is no such penalty for a fetus.

And why is there no punishment in terms of Jewish law for killing a fetus?

I do not know the laws of punishments. But it is likely that the Sages held that it is not a full person.

It seems to me that for a non-Jew there is a death penalty for this.

Correct, and that is a hint to the root of the law. It seems that the fetus is considered like a person, but not like a Jew. Therefore, the prohibition that applies is “whoever sheds the blood of a person” and not “you shall not murder,” exactly like a gentile. And from this it follows that one who kills it is not liable to death, just as one who murders a gentile is not liable to death, because there is no death penalty for “whoever sheds the blood of a person,” only for “you shall not murder.” (What I wrote was that there is a moral prohibition here even if not a halakhic one, but more precisely there is a halakhic prohibition here of “whoever sheds the blood of a person” and not of “you shall not murder”; rather, this halakhic prohibition is a moral prohibition like the other Noahide prohibitions.) But a gentile who transgresses “whoever sheds the blood of a person” is liable to death, and therefore that is also the law regarding a fetus. Study this carefully.

I liked that.