Currently, as I understand it (and as a cis person, my understanding may be flawed), the trans-friendly terminology uses "boy", "man", "girl" and "woman" to refer to psychological gender. Physiological sex is denoted by "male" and "female". So a trans man or trans boy is a man or boy with female sex traits, and a trans woman or trans girl is a woman or girl with male sex traits. (of course, the farther along in their transition, the fewer non-gender-matching sex traits they have)

Right, it's starting to make sense. I'm confused in particular about "MtF" and "FtM", what does that refer to? Since the M stands for "male" and the F stands for "female", which both refer to the physiological sex, is it used to refer to someone who has undergone the procedure to have the opposite sex's organ instead? Or is it used for someone who has begun HRT?

There really should be no reason to identify people based on their sex organs unless you intend to have sex with them. Referring to trans and cis men alike as men and trans and cis women alike as women, regardless of what is in their pants, is really all you need. If you need to refer to a person's sex organs for a legitimate reason, that person is a man or woman with a penis, or a man or woman with a vagina. (or a nonbinary person with a penis or vagina)

It was more about the scientific and educational aspects of it. You've already explained that bit, so.