This question connects to another question I once asked you about the expression “the general principle is” in the Mishnah in Berakhot, appearing at this link: https://mikyab.net/%D7%A9%D7%95%D7%AA/%D7%A7%D7%96%D7%95%D7%90%D7%99%D7%96%D7%9D-%D7%95%D7%9B%D7%9C%D7%9C%D7%99%D7%9D-%D7%91%D7%92%D7%9E%D7%A8%D7%90/

There you wrote: “The rule comes to include something additional that would not have been included without it.” If so, here too there appears a generalizing expression, “the common denominator,” which is supposed to include something additional that would not have been included without it (or perhaps one could answer that the expression “the common denominator” is not a generalizing expression at all). I’m not sure I understood your answer—did you mean to say that without the words “the common denominator,” I would not have been able to combine two primary categories into one subcategory?

Indeed, “this is the principle” comes to include something extra. But “the common denominator” comes only to say that one can derive a subcategory from two primary categories. True, one can ask that we would do this even without it, since the hermeneutic principle of the common denominator exists throughout the Torah, and so it is still difficult why this was written—but this can be pushed off:
1. According to the view of the Rosh there on 6a, this is not really a common denominator in the full sense, but a similar kind of derivation, since it has the defining character of pit (including exemption for vessels and for a person), and fire only helps remove the objection that another force is involved in it. And even according to the other view there, which holds that there are no exemptions at all, this is not truly a standard common-denominator derivation.
2. There was an initial assumption that in damages there is no subcategory actually derived from two primary categories (it could theoretically exist, but we have not found one. For example, regarding the Sabbath, we have not found a subcategory of two primary categories, except for spitting in the Jerusalem Talmud, brought by the Rema, which according to the explanation of Rabbi Meir Elyah cited in the Mishnah Berurah is winnowing and throwing together—and this is not the place to elaborate). And it teaches us that there is. This also explains why the Talmud there bothered to bring four different examples (and the medieval authorities already wrote that they do not disagree with one another). Each one demonstrates that there is another such subcategory.

As for your last question (whether this is a generalizing expression):
Rashi and the Rif disagree (see the novellae of Rabbi Yitzchak Zev HaLevi Soloveitchik on the laws of property damages) whether “the common denominator” in our Mishnah is the general primary category of damages, and all the primary categories were stated only for their specific laws, or whether each primary category is an independent category in its own right. That is, does “the common denominator” come to say that the obligating factor in property damages is one: your property, and its safeguarding is upon you (according to the Rif’s version, the Mishnah also includes “your property”)? And each primary category is stated only for its particular exemptions. Or rather, is each of the primary categories its own independent primary category of damage? The practical difference would be for something derived from two primary categories, or for something that resembles none of them but is your property and its safeguarding is upon you.
According to the view that this is one category, there is indeed a generalizing expression here, but that itself also explains why it is needed: to make clear that in the final analysis there is one general primary category of damage (and the cases on page 6 are practical ramifications). According to the view that these are several categories, this is not a generalizing expression; rather, it itself comes to introduce the cases on page 6.