The original description for this tag says that “One's Gender is something that can influence how others treat you. However there's more to it than just one's physical appearance.” Between this statement and the modifiers allowing one to distinguish "physical gender" and "mental gender", this tag is, by Internet standards, an early acknowledgement of the distinctions between what today we would call assigned sex and gender identity, though the standard “genders” seem mostly focused on physical characteristics. Wyrm's site notes that, not having been present in the original code, this tag's inclusion in the revised code was the result of a long and drawn-out discussion, so presumably some thought and consideration went into it.
That said, the language and public understanding of issues of sex and gender have changed a lot in the last twenty-plus years. A Dragon Code designed in the modern day would probably have a very different-looking tag, or probably several tags, with different subtypes and updated terminology, allowing finer distinctions between assigned sex, gender identity, gender expression, sexual and romantic orientation, etc.
Also, keep in mind that since the Revised Dragon Code was designed to allow description of not only nonhuman beings, but beings that do not make any appearance in our daily reality, how each term translates into actual features may vary; if there is such a thing as a male or female sentient cloud, for example, what that looks like would probably be considerably different than anything we're used to. Or, then again, maybe not.
In humans, someone assigned female at birth (AFAB) is usually someone with XX chromosomes and equipped with a uterus and ovaries and, as an adult, boobs. In other species, your mileage may vary.
Though this is still a scientific term used to describe organisms that can produce both male and female gametes, it seems likely that this was meant in its earlier sense to describe people we would now describe as intersex: having sexual characteristics associated with both males and females.
The “mental gender” version of this would, in modern terms, be a variant of non-binary gender identity, such as bigender.
In humans, someone assigned male at birth (AMAB) is usually someone with XY chromosomes and with dangly bits between their legs and, in adulthood, a tendency toward extra facial hair. In other species, your mileage may vary.
As opposed to someone who is intersex and would have sexual characteristics associated with both males and females, it can be assumed that a neuter individual would have either no sexual characteristics at all, or at least none that are clearly distinguishable.
The “mental gender” version of this would probably be a variant of non-binary gender identity, such as neutrois or agender.
Wyrm's site describes this term referring to “One who has some characteristics of both male and female, but is not fully hermaphrodite, e.g. feminising testicular syndrome).”
Like “hermaphrodite”, this has a scientific use but its intended meaning here is increasingly considered obsolete: it refers to someone who has a mix of sexual characteristics, but a majority of them are associated with one sex or the other. Put another way: if one considers percentages of physical sexual characteristics on a spectrum, with “male” at one end, “female” at another, and the midpoint consisting of “hermaphrodites” as those with an equal mix, “pseudo-hermaphrodites” would occupy the intermediate points. In modern terms, it would probably be much more appropriate to describe people in this condition as a variety of intersex.
The “mental gender” version of this would, in modern terms, be a variant of non-binary gender identity, such as demigender.
Wyrm's site notes, “If you want to specify the various genders, include them in parenthesis after the ~. For example, G~(fm) indicates: 'I vary between female and male.'”
Given the apparent physical orientation of the other subtypes, this is probably meant to describe someone who can change physical sex - and (given the separate modifier for transitioning between specific genders) probably meant to indicate someone who can or will do so on short enough time scales that indicating the exact state of transition in one's Dragon Code would require an infeasible frequency of changes.
The “mental gender” version of this would probably be a variant of non-binary gender identity, such as genderfluid.
Possibly an alternative to neuter, or possibly just someone who feels no need to be definite.
The “mental gender” version of this would probably be a variant of non-binary gender identity, such as xenogender.
Used to indicate someone undergoing a change of sexual characteristics, presumably at a slower speed, or with a more permanent implied outcome, than indicated by the variable gender subtype, with the beginning and the end states listed on either side of the symbol, ex. Gm>n for “I am a male who is changing to a neuter.” There is no explicit distinction here between a real-world transition, involving things like hormone replacement and gender-affirming surgery, and some kind of shapeshifting.
Though the sources do not specifically mention the use of this modifier with “mental gender”, in principle there's no reason why it couldn't be, provided that there was a standard order of operations between > and /; consider Gm>f/h which could mean something like either “I am physically male transitioning to physically female, but I identify as non-binary” or “I am physically male but my gender identity is shifting from female to non-binary”.
Someone who wishes to define their gender in terms other than those foreseen by the tag can put it within double quotes after the tag header, ex. G“sex”.
Someone who wished to specify a gender identity different from the sex assignment implied by their bodily characteristics could indicate it using this modifier with the “mental gender” - their gender identity, in more modern terms - after it, as in Gm/f for “I am a female trapped in a male's body”.