The Mishnayos of Bava Kama are written in a unique style. You will notice this in a number of places in Bava Kama.

I would like to suggest the following answer to your specific question.

1) The Gemara earlier (6b) states that the words in the Mishnah (2a) "Chav ha'Mazik" were said by a "Tana Yerushalmi" who spoke in a concise way.

It seems that the Mishnah (9b) is also stated in this concise style. Speaking in the first person is also a more direct, possibly we could say colloquial, way of speaking, and is suited to the way of speaking in Yerushalayim.

2) I also found that the Mishnah in Kidushin 28b states, "His speaking to the high (i.e., to the Beis ha'Mikdash) is equivalent to his giving over to a person." However, I found in the Or Yakov text of the Talmud Yerushalmi that it says, " My speaking to the high is equivalent to my giving over to a person." Again, we see that the Yerushalmi style is in the first person. (This variant text is also mentioned in the standard Mishnayos in Kidushin in Shinuy Nuscha'os).

3) We also find other Mishnayos in the first person. See Shevi'is 2:3 in the name of Rebbi Shimon, and see Shabbos 150a in the Mishnah in the name of Abba Shaul. It may be that this was the older way of speaking in the time of the Mishnayos.

4) I found support, bs'd, for some of what I wrote above from the Mishnayos in the final chapter of Maseches Kesuvos. The first Mishnah of the 13th chapter of Kesuvos tells us that there were two "Judges of Decrees" (Dayanei Gezeiros) in Jerusalem. One of them was Admon. In Mishnah 3 we find Admon speaking in the first person ("Why do I have to lose out because I am a male?!"). In Mishnah 9 we also find Admon speaking in the first person ("If I owe you money why did you borrow from me?!"). Shinuy Nuscha'os (printed in the margin of the Mishnayos) writes that Admon's style of speaking is unusual, since he speaks in the first person, in the way of the second Mishnah in the first chapter of Bava Kama.

5) I can now suggest why Admon speaks in a similar way to the Mishnah in Bava Kama 9b. As we saw above, the Mishnah on 2a was taught by a Jerusalem Tana. The Maharatz Chayus to Bava Kama 6b writes that the first 3 Mishnayos of Bava Kama are different from other Mishnayos because they were said by a Jerusalem Tana. We can now add that this is why they are said in the first person, because this is the "lighter" style mentioned in Bava Kama 6b. And this is why Admon also spoke in this style, since he was also from Yerushalayim, as we learned that he was a Dayan in Jerusalem.

6) In addition, I checked the standard editions of the Talmud Yerushalmi, and every time that the Yerushalmi cites the Mishnah in Talmud Bavli in Kidushin 28b (" his speaking to the High"), the text is " my speaking to the High." Again, the Talmud Yerushalmi speaks in the first person, because it was written in Yerushalayim.

7) I later found two additional explanations in the commentaries:

a) "Kol ha'ReMeZ" is a commentary on the Mishnah by Rav Moshe Zacuto (1625-1697, born in Amsterdam; he was from the same family as Rav Avraham Zacuto, 1452-1515, a Spanish astronomer, who in a much earlier period invented the astrolabe and astronomic tables which helped Columbus on his journey of discovery to America). Kol ha'Remez explains our Mishnah based on the Mishnah below (26a) that says "Adam Mu'ad l'Olam." A person is always liable for any damage he does, whether it happened deliberately or accidentally. This is because, when it comes to damages, everyone is like a "Chaver," a Talmid Chacham, who does not require a warning to make him liable (see Makos 6b). Our Mishnah speaks in the first person to hint that everyone is equal when it comes to his responsibility to guard his property.

b) The Sefer "Ahavah b'Ta'anugim" (literally, "Love with Delights," because learning Torah is the greatest pleasure in the world) was written by Rav Avraham Azulai (the great-grandfather of the Chida) who lived between 1570-1643 and was the Rav of Hebron. The Ahavah b'Ta'anugim writes that our Mishnah is written in the first person to teach that the laws of damages are logical ones. They do not belong in the category of "Chukim," the laws of the Torah whose reasons we do not always understand. They are "Mishpatim." Everyone agrees that he has to make himself obligated for any damage he does. Even if we did not possess the Torah, a person's "Sechel" would be sufficient to know that he has to pay for the results of his actions. The Mishnah is written in the first person because every individual would admit that "I am liable for what I did."

Dovid Bloom