Let me give you a clue: nobody cares about psychologists. This is why when somebody says they want to become one, many people roll their eyes. Because there are a lot of things they say that are nonsense. I am not saying that "man" does not have other connotations for "man" - but I do not care what people subconsciously think I when they hear this word. Only you do - because it gives you an answer that satisfies you. ALL you have to do is ask people their definition of "man" and "woman". That is how a word is defined. When anybody hears "Look, there's that man we met yesterday" - what does a person expect to see? If they turn to see somebody they know identifies as a man in a "female body", they aren't going to be very impressed with whoever said it.
If somebody wants to point out or talk about a person that is clearly transgender (let's say a man, with a man's body, who has perhaps taken female hormones and maybe even had breast implants) the person will say "did you see that man over there?" If they know the person, most likely they'll call them whatever they want to hear - I know I wouldn't call somebody otherwise, even if I don't believe it.
You keep saying about what psychological studies have shown how people use the word for. Guess what, when it comes to the definition of normal words, nobody could care less about your studies. In the average population that is. They impress nobody, I'd roll my eyes. Go up to someone in the street, ask them their definition of "man", hear them say "an adult male" or something along those lines ("has a penis/man's body etc.), then tell them that they are wrong, they don't define "man" that way, because you have psychological studies that prove otherwise. Many people would tell you to f' off.
If you are so sure about your definition of "man" and "woman", why don't you contact some dictionaries, and explain your case to them? If you do look it up in a dictionary, you will find quite a few will be found, such as "Man - referring to the human species 'A great leap for mankind'". You will not find any listing for a gender identity. Let alone it being the primary definition! Of course, you are in denile, deluding yourself by taking psychological studies and using them to define a word, so inevitably you'll dismiss this.
I can go back to normal life, with my friends, average people, and know that if I ask them what a "man" is, or if they say "that man" We both know what we mean. There is an understanding. I'm perfectly happy with that despite knowing a very small minority thinks otherwise, and that should we meet a trans person we'll ignore our beliefs for the sake of politeness and not offending. Can you and the others here honestly say that what the average population would define "man", if you asked them, as an adult male, you'd be okay with that? You'd contentedly think "Oh well, the majority of people have been wrong before. They're just like people who thought the Earth was flat." I think you wish that the public agreed with your definition. But they don't. Just accept that. You can carry on believing that people who identify as a man in female bodies are men, but if the vast majority of people in society would not think of them truthfully as a man, then there is no true recognition of them as a man - only by a minority. In society's eyes they are not. Just look at my first post on this thread:
"I define being male as possessing male genitalia. If you were born with it, then you were born male. Simple as that." (paraphrased)
This received 5 positive reps - clearly a lot of people in society would also agree.
When you might be talking to somebody about a trans person, and you say "he is trans", do you know what people are truly thinking? They know that you refer to them as a "he", but inside their head, what they want to know is if he has a male/female body. If they were confident enough to ask you, they might even say: "So, is he a man or a woman? Did he have a sex change from a woman to a man?" You would probably say your spiel about "man" referring to gender identity, but you see, whilst it might be hurtful for trans people to her, most people just want to know whether somebody has a male or female body to decide whether they are a man or woman. This is why you will hear people say "is that a man or a woman" - in your terms, what they mean is "are they male or female" - but there is no important difference to them. You are so desperate to believe otherwise that you believe whatever you want to believe. But don't define a word by psychological studies, that is just foolish.