I meant this one (and in fact the terms here are intention and awareness): Chiddushei Rabbi Chaim HaLevi, Laws of Levirate Marriage and Halitzah, chapter 4:
“And it seems this proves necessarily that the law of divorce and betrothal differs from the law of halitzah and the requirement of being for its own sake, for in divorce and betrothal one needs the intention of the owners regarding the very divorce and betrothal, because the essence of betrothal and divorce is effected by the owners, and they are the ones who prohibit and they are the ones who permit, and therefore their awareness is required, and it does not help that others instruct them, since they themselves are not mentally competent. But in halitzah and in the requirement that it be for its own sake, there is no law of the owners’ awareness; rather, all that is required is intention regarding the act of halitzah itself, and the exemption comes automatically by Torah law. This is what Tosafot brought: that where only intention alone is needed, it helps if others instruct them, just as it helps to establish that it is done for its own sake; and there too we require only intention alone.”
And in Oneg Yom Tov:
“And the reason for this matter seems to me to be that this works specifically with regard to the scribe’s writing, where guidance while standing over him is effective. But regarding betrothal and divorce, which depend on the husband himself, and similarly regarding terumah, it is not effective. For in truth, in the thought that the Sages required together with the act, there are two kinds of thought. One is thought that directs the doing of the act, meaning that he must think in his mind why he is doing this act. The main effect of the matter is completed by the act; it is only that the Torah said that the act does not take effect unless at the time of doing it he thinks in his mind why he is doing this thing. And there is thought that itself causes and effects the matter of the act, as in the case of a bill of divorce and all acquisitions, where the woman is not divorced unless the husband wants it, and a sale is not acquired unless the seller and buyer want to transfer and acquire. It is only that the Torah said that the bill of divorce is not completed by words and desire alone unless the matter is done through an act, such as a bill of divorce through writing and giving a document of severance, and a sale through money or pulling. But in truth their desire also effects the acquisition, and it does not come merely to indicate the function of the act, why he is doing it; rather, it is part of the essence of the acquisition itself, except that this desire does not produce its effect except through an act.”

Desire describes an attraction / longing / craving for a state of affairs that a person thinks / believes / feels would be good for him if it were actually realized. “He fulfills the desire of those who fear Him,” “to do according to the wish of each and every man.”

Intention refers to the particular desire that lies behind performing certain acts.

There is another missing factor here, namely actual implementation. Very often there is desire but no action.

And the connection to knowledge is that in order to know which actions to take, one has to know how the world works.

Intention is from the root meaning direction. The direction is dictated at every moment by: desire + desire to implement + worldview.

Decisor, I’m afraid that here we’re dealing with halakhic concepts, and their interpretation needs to stay close to the contexts and implications.

So you didn’t mean that. I was speaking about Rabbi Chaim’s distinction in the laws of unintended acts.
Here it seems the distinction is about awareness. A deaf-mute, an imbecile, and a minor lack legal awareness, and therefore even teaching them doesn’t help. But where all that is needed is intention for the act itself (so that it not be done as mere uncomprehending activity), without understanding what he is saying, then you can also teach them and that works.
In halitzah there are several proofs that full legal resolve is not required; rather, the act simply has to be performed by the person. But divorce and betrothal are acts that create legal status, like ownership, and there awareness is required.

Cardingo, desire and intention are not halakhic concepts but human psychological concepts.
And Jewish law tries to clarify what they are and what to do with them.

Thanks

It’s somewhat like what Rabbi Shimon Shkop mentions in several places: that there is intention regarding the act — that I intend for the act to be done — and there is intention regarding the legal effect, that the legal effect should take place. Rabbi Chaim just calls it by more general terms.

Exactly right

One is the opposite of uncomprehending involvement, and the other is the opposite of an unintended act.