Actions: Intentional Under Some Description? [Correction]

In an earlier post on this topic, I mis-attributed the origin of "intentional under a description" talk to Davidson's 1963 paper, rather than Anscombe's 1957 book Intention. More importantly, I wrote the post without having read Anscombe's excellent 1979 paper "Under a Description", which clears up a number of things about what is supposed to be going on with the view. In the next few days, I hope to have a post up detailing my new and improved understanding of this "under a description" talk. I know see ways in which my presentation of the view in the previous post were missing what is, essentially, the key element of the view (hint: my statement of component (B) of Davidson's view in the earlier post is thoroughly incorrect).

Actions: Intentional under some description?

This post is subject to an important correction.
I think the origin of talking about actions "under a description" is Donald Davidson's 1963 "Actions, Reasons and Causes". If I am right, Davidson does not so much provide an argument for his thesis, but rather, simply puts it forward:
"I flip the switch, turn on the light, and illuminate the room. Unbeknownst to me I also alert a prowler to the fact that I am at home. Here I need not have done four things, but only one, of which four descriptions have been given.[...]Since reasons may rationalize what someone does when it is described in one way and not when it is described in another, we cannot treat what was done simply as a term in sentences like 'My reason for flipping the switch was that I wanted to turn on the light'; otherwise we would be forced to conclude from the fact that flipping the switch was identical with alerting the prowler, that my reason for alerting the prowler was that I wanted to turn on the light."

It seems that Davidson's view here has two components:
A) My flipping the switch = my turning on the light = my illuminating the room = my alerting the burglar.
B) My flipping the switch was intentional, though my alerting the burglar was not.

Here is a reason to think that if my φing = my ψing, it is not possible that one was intentional and the other not:
1) If my φing = my ψing, then for any property P, if my φing has property P, so does my ψing.
2) Suppose that my φing was intentional but my ψing was not.
3) Then, there is a property P (namely: being intentional) such that my φing has P, but my ψing does not.
4) Then, it is not the case that my φing = my ψing.
5) So, if my φing = my ψing, it is not the case that my φing was intentional while my ψing was not.

Granted, Davidson, it seems, would either deny (1) or the inference to (3) under the supposition of (2). I would have thought that (1) is an uncontroversial instance of Leibniz's law, so I assume it is more likely for one to deny (3). But, at the same time, being intentional seems like a perfectly nice property.

Since I think we should concede component (B) of Davidson's position, I can only assume the motivation to reject (1) or (3) in my argument comes from some good reasons to accept component (A) of Davidson's position, however, it seems like we also have good reason to abandon (A):
1) My flipping the switch could have occurred without the light being turned on.
2) My turning on the light could not have occurred without the light being turned on.
3) So, my flipping the switch is not the same thing as my turning on the light.
(repeat with the necessary alterations for each of the items being identified).

Here's a naive conclusion to draw from my two arguments: My flipping the switch is not the same thing as my alerting the burglar, and thus, we need not appeal to the notion of an action's being "intentional under some description" to explain how my flipping the switch is intentional when my alerting the burglar is not.

But perhaps I am being insufficiently charitable to the Davidsonian position.