And why is it really detached from the primary category and defined by the act rather than by the result?
And how can we derive from a primary category that is result-based to a derivative that is act-based?

I wrote this already. Sometimes part of the primary labor is defined as a derivative. See making the “eye” of the dye, in Maimonides, Laws of Sabbath 9:14, and elsewhere. Similarly in Maimonides at the beginning of the Laws of Leaven, where sourdough is considered leavened food (even though it is not fit for eating) because dough is caused to ferment with it. Making a tent is a derivative of building, and so is making cheese. Each of these is a partial act of the primary category: creating a space and gathering parts together. See my columns on the labor of building.
As for the question of how this is learned from the primary category, that depends on how you understand primary categories and derivatives of labor on the Sabbath. According to the approach that every significant labor is forbidden, and the classification into primary and derivative categories is according to the Tabernacle, it is enough that wringing is significant. It just was not established as a primary category because it is included within laundering.
By the way, you do not have to see this as an act, as opposed to washing or cooking. All of these are acts that can be defined by their result. Even carrying is not necessarily defined as an act, but by its result.

I agree that if this is the conclusion of your words, it is proven from the Talmudic discussion.
But the question is how it is built on the primary category.

I explained. Because it is part of it. Wringing is part of the laundering process.

According to this assumption, that a primary category is what existed in the Tabernacle and every significant thing is forbidden,
doesn’t significance have to be based on the result?
And even if the whole definition is, as you wrote, that it is one of the needs of laundering — and Maimonides says so too —
doesn’t it need to be part of laundering in the sense of producing cleanliness as a result? Or at least as part of a cleaning process?
And really, why is it significant if it is not doing any cleaning?
And you wrote, “By the way, you do not have to see this as an act, as opposed to washing or cooking. All of these are acts that can be defined by their result. Even carrying is not necessarily defined as an act, but by its result.”
I didn’t understand what you meant. Does that mean this too has some defined result? Or that there one can define a result and here one cannot?

We’re repeating ourselves. I explained.
What I mean is that there is no necessity to define wringing as a pure act-prohibition. It is forbidden to do an act whose result is draining out liquid.

I understand.
And according to this, it comes out that wringing does not have a uniform definition with laundering, which is removal of dirt, or restoring the appearance to what it was before.
Rather, wringing is part of the laundering process, but it is the draining of liquid.
That is on the assumption that a primary category and a derivative do not have to share one identical definition, but rather the same basic idea or action-name.
Is there a side that says they do need the same definition?

Correct. What do you mean, “is there a side”? You can see here that this is not so. Sometimes the derivative has the same result and sometimes not.

Even within laundering itself, some say that there is washing even with a clean garment; regarding soaking, soaking it is its washing.
Like in the case of saving a garment from a fire by putting water on it — Rashbam’s view is that this is forbidden.
Is there a way to understand that — what makes it different from cooking?
Unlike wringing, where I understand what you wrote.

That view is not acceptable to me. Unless you define soaking — as we did regarding wringing — as a derivative that is also an element of the laundering process.

With wringing I can understand that since it is an element in laundering, and it always applies to a garment that is already clean, after laundering, and it is part of the process of making the garment ready, it is a derivative — because draining the water hastens making the garment ready to wear, and it has a benefit even for a clean garment, since once it has water in it, it speeds the drying.
But with soaking a clean garment, he did nothing at all (and even caused damage), so why should that be considered laundering?
And to say that it is doing laundering even though there is no result, and only the act itself is forbidden — why is that really different from the other labors, where we do not say that the act in and of itself is the labor, but rather the result?
And it seems more reasonable to say that this is a forced approach.

I said that on the face of it this is a forced approach. There is no obligation to reconcile every view.
But what I also wrote is that soaking, as a derivative, is not necessarily part of laundering. Ordinary laundering includes both soaking and wringing. Now wringing and soaking have been forbidden as derivatives in their own right, and now there is no necessity that either one be done for the sake of cleaning the garment. The act itself (or the result) is forbidden.