We're closer, to be sure, and I agree with much of what you said. There is still more than a semantic difference, however. To get around that, let me refine my terms a bit more.
First, "identity" is a term that's used broadly and when so used, as you indicate, connotes both things core as well as constructed ("forged"). "Gender identity" is an unfortunate term, because gender itself is an overloaded term. What I mean by identity aren't the things most people force into "gender" at all. Rather, because I see transsexuality as an intersex condition, the core of which we are speaking reduces to a body mapping instability. Not pure male or female in the cis sense, but a compound of both. There is a regrettable tendency in western culture to speak of the psyche as the reality and the physical as the illusion or overlay. But in fact we actually ARE both, compounded into something unique. Not even 1 + 1 = 2, but simply ... 2.
That we can look at the condition from partial perspectives helps us to treat the psychological conflict that arises, but it's important to note that the condition doesn't really change. SRS, FFS, hormones, etc. effectively mask the body mapping problem trigger, but still leave us (MtF) physically male in most ways. In essence, we resolve instability by introducing a countervailing imbalance. (I hope to be unbalanced soon ... LOL!)
Gender Dysphoria - real GD (see footnote), that is, not the usual stuff that passes for GD, is the only real manifestation of the condition itself, i.e., of the real, or combined "identity." That's not to say one can't understand their psyche as female, but again, THAT realization is lizard brain stuff. There are no aspects of the heart, of feelings and emotions, not even widely acknowledged sexually dimorphic biases and tendencies that are determinant that one's psyche is male or female in any given case. At some point, one simply knows whether their psyche is male or female. That can be apparent from early childhood or resolve later.
I harp on this because of the problems people have distinguishing and conflating being cross-sexed (transsexual) versus being gender variant. The most egregious result of using terms in the emotional space as it applies to transsexuality is that it encourages the spectrum concept and fosters the problem of the gender variant thinking they "identify" with trans women because of what they "feel." A friend recently put it this way (paraphrased): "These are people who are flirting with their femme side. Their saying the words IS their femininity."
(footnote) I would recommend a careful reading of the DSM 5, including both the diagnostic criteria AND commentary for Gender Dysphoria. Psychological fallout and various kinds of clinically significant distress aren't GD, they are part of the criteria FOR GD.
