I’d be curious whether it’s even that distinct from how ‘empathy’ commonly works in practice.
I’ve run into a fair number of basically decent sorts, who think of themselves as empathetic and care about others; but whose practice of ‘empathy’ is basically just projecting themselves into the other person’s situation and then treating them as they would wish to be treated were they in that situation according to how that situation would make them feel.
It’s clearly not an ‘asshole version’ of empathy; and it’s even a pretty adequate substitute for actually being empathetic in the theoretical sense if you are dealing with others who fall into basically the same region of approximate psychological normalcy as you do; since using how you would feel as a proxy for how someone else feels is pretty accurate if they feel much as you do; but it quite dramatically turns from looking like empathy to looking like distressed-and-well-intentioned confusion when they run into a case that really isn’t them.
If you are just projecting and you are a dramatically bad or weird or both person in one way or another the fact that you actually have extremely weak ability to discern or model other people’s states of mind will become evident more or less swiftly and dramatically; because using the heuristic of assuming that everyone else is as awful as you is blatantly contrafactual and will lead you to absurd results in short order; so projection certainly seems to be most visible in various flavors of egregious asshole or weird psych case; but there is also the case of the mostly benevolent and mostly normal projector who typically passes for empathetic because they really do care and are ordinary enough that just assuming that others must feel as they would does not lead to absurd conclusions in the bulk of ordinary cases; but leads to wildly uneven performance between, say, ‘empathizing’ with a friend who is worried about what one of their children has been getting up to in a way that’s basically the same as how they would worry about one of their own getting up to that; and just hitting a blank wall of visible incomprehension when trying to help a relative who has been doing some hoarding and could probably use a professional look at those anxiety behaviors; or attempting to relate to someone with an autism spectrum disorder that they do not have.