Most of us don't tend to like the language "male at birth" or "born male" because it implies that we don't know who we are, and that we chose to be trans (we didn't). It's generally better to say assigned male at birth, or assigned female at birth (acronyms of amab or afab, there are a few others, but those are the ones you'll see the most). Some older transgender (or transsexual, as some older trans people still identify with that word - although most of us reject it now since it puts emphasis on whether we've had "the surgery" or not) people may still use "born male" or "born female", but that's rarer. The new language is more inclusive, doesn't invalidate, and includes intersex people as well.

A transgender man is a man. Period. That means he was assigned female at birth - not male - and transitioned, is transitioning, or identifies as male. It's important to note that not all of us do transition. Some of us only transition through hormones (estrogen or testosterone), some of us only through surgery, some of us through both, some of us only socially (as in we do neither, but do use our new names, present as our gender, etc), and some choose not to transition at all for whatever reason (safety being most common). We are all transgender, though. A lot of the memes floating around about the bathroom issue neglects that fact, as they only validate the experience of transitioned transgender people. This gives the illusion that only those who have transitioned are transgender... which erases the rest of us.

Assigned male at birth would be a transgender woman, but could also include genderqueer, non-binary, gender fluid, etc. I'll let someone with more experience on those identities cover that, though... but a transwoman was usually assigned male at birth.

We could say just man and woman to some trans people. But you have to understand that for many of us, this is a huge part of our identity. As someone who will likely never transition medically, it is important for me to identify as trans. This is because I will never "pass" (I do hate that word, but it's hard to find any other fitting one) well enough to fit in with cis men. And I have never, will never fit in with women. Thus, I tend to gravitate toward LGBT people, and fit in the most among transgender people who have - also - not transitioned. Some trans people choose to go stealth after transitioning, and these trans people may reject the trans part of man or woman, and simply prefer just man or woman. For a lot of us, though, it's literally who we are. You can ignore the label, but it won't erase the stigma, harassment, dysphoria, etc that we face day to day. The label matters.

ETA: I realize, too, that many of us hate the word "identify as", but in that context I wasn't sure how else to word it. My apologies.