The concept you describe is indeed important and should be understood, absolutely, but don't blame the reader for the authors failure to communicate. Especially when that failure is mostly intentional as part of a struggle to redefine a word that has significant power. A very useful tactic when it succeeds, but that battle has been lost in this case for quite a while.
Personally I'm quite happy with the common definition of "racism" because the belief you describe is too tied up with the dated concept of "race" (which doesn't exist in reality) and means that the determining factor in whether an action is "racist" is the skin colour and ancestry of those involved, neither of which is under the control of any participants. The other definition means it is determined by asking if the action was prejudiced and if that prejudice was related to race.
There is a very good point to make here that the more common definition can make it more difficult for people to understand the effects of generations of oppression can have on a class of people, even on that society generally considers overt racial prejudice to be unacceptable. The definition you describe does not have this flaw.
For what it's worth, if I am treated in a prejudiced manner by person A, and I treat person B in a prejudiced manner, and the only thing connecting A to B is race, that too is racism.
Or in simpler terms, two wrongs don't make a right.
One could argue it gives minorities license to say as many racist things as they want to and claim it isn't racist. It's an argument constructed so that one side wins by default.
"Minorities saying racist things", by this theory, doesn't hold a candle to, say, the fact that minorities are more frequently imprisoned, more often pulled over/searched illegally/beaten by law enforcement agents, more often feared, more often treated with bias in the workplace or dating world, and on and on and on and on and on. Racism isn't "the act of saying something bigoted", it's the acknowledgement of a systemic bias based on a person's skin color. And it doesn't deny that bigotry exists in both directions – it just insists that to focus on bigotry is to miss the deeper issue.
This is relevant to this thread, because gender inequality works much the same way. What Adria did was a dick move, and I think her actions were and continue to be rude as hell. But she acts within a greater context of sexism and misogyny in the programming world – hell, in America at large – and to completely disregard how shitty an environment exists for many women is to miss that Adria's behavior stems from a crappy reality. Doesn't make what she did any less inappropriate, perhaps, but it suggests that maybe there are issues here more important than what she did which may be more worth focusing on. I mean, any ills resulting from her behavior have been more than counteracted by the death and rape threats that have been fired her way – she may have started this, but the backlash against her has been obscenely disproportionate.
I think having a definition of racism to be "discriminate against somebody else due to skin colour", as well as a notion of "institutional racism", which is the same thing but on a systemic level, would make more sense than the more academic definition of racism.
Racial hatred is despicable in all forms. Those who push for such an artificial redefinition of racism would have us forget this.
Therefore, no, she cannot expect her audience to adhere to her particular definition of racism since it is highly non-standard outside of certain limited circles.
For what it's worth, I also happen to think her "definition" of racism is rubbish. It makes all kinds of assumptions about power dynamics in a given situation and injects motivations into actors where none are known.
Presumably Adria is claiming to be without power. She has the power to get someone fired with a single tweet. She has so much power she's gone mad with it.
I see people post this a lot without understanding the point of what she's saying. Adria subscribes to the belief that "racism = prejudice + power", which is basically to say that, yes, black people can be hateful towards white people and yes, racial discrimination does occur within the black community, but RACISM is the result of systematic oppression over hundreds of years.
The reason this distinction is made is because black people discriminating against white people is a response to the racism that they face every day, which is very different from white people thinking that blacks are inherently inferior and should have their rights stripped from them.
Please, educate yourselves on what Adria's saying. I know it all seems so foreign, but beyond the fog, there is a very real and very true message that should be heard and studied.