Here is your argument: If you say "The easiest way to launch an undefendable argument against a person who is a brown elephant from mars is to claim that they have too few stomachs" and then I accuse you of having too few stomachs, obviously I am wrong, or there is something inherently wrong with my argument. This is false.

In addition, the fact that she was accused of being too emotional is supposed to a) only apply to women b) be a bogus argument in general and c) indicate sexism.

I know a) and b) are false because I have seen tons of men being too emotional and this causing real damage. A very very common example is when a product that has tons of hours, blood, sweat and tears invested into it has to be tossed because of strategic reasons. Employee X will not have any of that and will relentlessly lobby and defend the product, because the product is their baby. X is being too emotional.

In the author's defense, it didn't seem to me that the author was being too emotional in the incident. From the information provided, that is a false accusation.

About c), nothing in the story as it is seems to indicate sexism to me. Make no mistake, it looks like an unfair attack on the author. Maybe everyone hated her for some other reason, or maybe it was really sexist. We, as readers, do not know. Maybe there were other elements to the story, other events / conversations / attitudes that did not make it to the post. Who knows? If those did exist and were in there, maybe the sexism argument would be valid. Being unfair towards a person who happens to be a woman is not the same thing as sexism. The latter is a subset of the former.

When everyone in the comments accuse the author of being too emotional, they are probably referring to the fact that she is telling a story where sexism is not apparent (from the given evidence) and she is immediately attributing it to sexism.

This is a feared situation. People are afraid when someone who is in the abused side of an anything-ism call other people abusers without apparent reason. For example, I am middle eastern, and it's scary to other people when I start calling people who I don't get along with racists. This is similar to the sort of trump card argument that the author hates. It villifies the alleged abuser, even if they were not guilty, and the alleged abuser is in a disadvantaged situation to properly defend themselves. If I do this, inadvertently or not, this is considered a low blow.

You will see people all around you bending over backwards to avoid this sort of dynamic: Sheltered white people may act all nervous and awkward around people of other races because they try to act nice but come off overly fake, german people may do the same when conversation is started about jewish people (as if it was them, not other people, that caused the holocaust or as if they were guilty by association) and hell, even some MIT students will not say where they graduated from unless specifically prompted, because they are afraid to be percieved as the stereotype of high and mighty jackasses and not fairly as themselves.

The point is, sexism is a serious accusation that is not to be thrown around lightly these days. If it is to be made, there needs to be serious evidence. Otherwise, it's just foul play. Unfair attitude towards a person who is a woman is not the same as sexism.