As a native German speaker I believe I can expand upon, and slightly disagree with, your definition.
I suspect that a significant portion of the misunderstanding about slave morality comes from the fact that the german word "Moral" (which is part of the Netzschean-term "Sklavenmoral") has two possible meanings, depending on context: Morality and morale, and it is the latter which I consider to be the more apt translation in this case.
Nietzsche was really speaking about slave morale. It is important to understand that slave morality is not an ethical system or a set of values, rather it is a mindset which facilitates by psychological mechanism the adoption of certain values and moral systems.
To be more concrete, it is a mindset that Nietzsche suspects is common among the downtrodden, raped, unlucky, unworthy, pathethic, and unfit.
Such people, according to Nietzsche, value kindness, "goodness of the heart", humility, patience, softness, and other such things, and tend to be suspicious of power, greatness, risk, boldness, ruthlessness, etc.
To the slave, the warmhearted motherly figure who cares about lost puppies is a perfect example of what a good person is like - in sharp contrast to an entrepeneurial, risk-taking type of person who wants to colonize the universe or create a great empire or whatever.
To the slave, that which causes fear is evil - to the master, inspiring fear (or, rather, awe) is an almost necessary attribute of something great, worthy, good.
So, returning to your definition: Slave morality gives rise to the idea that he who is a good boy and cleans his room deserves a cookie. That, I would agree, is a significant consequence of slave morality, but it is not its definition.
I think Nietzsche would agree that "slave morality" originated with Jesus. The main new idea that Jesus brought as a moral philosopher was compassion, feeling for the other person. It's pretty find to hard in earlier sources, for example the heroes of the Iliad hurt weaker people without a second thought.
To me it feels obvious that the idea of compassion needs to exist, and needs to have force. Because otherwise we'd have a human society operating by the laws of the natural world, and if you look at what animals do to each other, there's no limit to how bad things can get.
Can compassion also become a tool of power and abuse? Sure. But let's not go back to a world without compassion, please.
This mindset has a failure mode of no longer being sensitive to the oughts and only noticing descriptive facts about the world.
Another thought on this is that people often talk about smothering social norms or moralities by giving examples of petty, infantilizing rewards for behaving in infantilized ways. What about a libertarian-oriented society that enforces rules primarily intended to prevent people from restricting one another's behavior? What about an economic system that encourages entrepreneurship and invention by doing things like enforcing contracts, or providing a social safety net so that more people can start businesses or create products without worrying that they'll literally starve to death in the street if their first idea doesn't work? Participants in a society like that should still maintain some awareness that this is the work of humanity, not an expression of a fundamentally just universe, but it's hard for me to argue that just participating makes them into slaves.
This description does a good job of providing two kinds of evocative theme but I think it doesn't draw out the connections or distinctions that need to be clarified when people are interacting with is vs ought, or perhaps with cosmos vs. society. When describing everyday life as a physical object in the universe, I think a rock-bottom existentialism is obviously right: The universe does not owe you anything. God isn't going to punish your oppressors or reward you for being good, and God also isn't going to punish you if you get what you want by being arrogant and doing things that are shocking. If that's master morality, then fine.
But when we form a society, we can choose to make a world where people who follow the rules are rewarded. We can describe obligations for people to participate and belong, and actively put energy into the system to buffer those people from random shocks. I can see how that reads as slave morality from the perspective of someone who's an unreflective participant in it, but I think that's also an unsatisfying way to describe an active project to defy the nature of the universe and, through work, create a world more to our liking.
So in sum I think your description here needs to clarify when this kind of be-good-and-you'll-get-a-cookie living is bad simply because it's inaccurate (God is not going to give you a cookie) and when you think it's wrong because, even if people establish an enclave in which rules are enforced and people are taken care of, that diminishes the human spirit or is actively harmful to its participants.
It might also just be a matter of hierarchical perception. To use a smaller scale example:
Lately, there has been a discussion between Bentham's Bulldog, Scott Alexander and others about the nature of the slave morality. The outcome for me is that I no longer have any clue about what people meant by it, or, for that matter, what Nietzche once meant by it. To sow even more confusion, here's my interpretation, which, I think, is different from those proposed so far.
Slave morality is a child-like assumption that when you clean up your room, you'll get a cookie. An assumption, that if you follow the rules you are entitled to some kind of reward.
The opposite, the master morality, says that whatever you do, you are not entitled to anything. The world doesn’t care. It's all up to you.
A nice and heart-warming example of master morality comes from American Declaration of Independence with its "life & liberty & pursuit of happiness".
It does not guarantee happiness, however modest. It only guarantees the freedom to give it a try. If you fail and end up unhappy, too bad for you, Declaration of Independence has nothing to console you.
Wikipedia lists comparable mottos worldwide. Say "life, liberty, security of the person" or "life, liberty, enjoyment of property". And you can see how those are a kind of anti-climax. Where the Declaraton of Independence boldly states that everyone is free to pursue happiness in their own way and leaves the resolution of the necessarily resulting conflicts for later, the alternatives give up on master morality and instead try to provide some kind of petty mundane guarantees, only suited to satisfy a slave.
As for slave mentality, a nice example comes from the book "Dune" by Frank Herbert. If you happen to not know, it's a science fiction story in which the humanity lives in a feudal empire spanning thousands planets, with an emperor, noble houses, peons and all the other feudal paraphernalia.
The empire is based on faufreluches social system . The motto of the system is: "A place for every man and every man in his place."
In the Dune world there's no pursuit of happiness. No sir, there isn't. But what you get instead is a guarantee that, as long as you fulfill your obligations, you are not going to be left alone. There's a you-shaped hole in the society and you neatly fit into it.
It's not the same heart-warming feeling as the one evoked by Declaration of Independence, but a heart-warming feeling nonetheless. Right-wingers dream about it, imagining the rural past where everyone was a part of village community, had their place and their obligations and was cared for by the others. Centrists have their welfare state. Left-wingers have their communes.
(A reddit commenter also helpfully notes that the faufreluches motto comes from the Fifth Annual Report of the Board of Managers of the Prison Discipline Society, Boston, 1830.)
Now that I think of it, the biblical story of Job is a morality play about slave mentality.
"God!" says Job: "I've been good. I've fulfilled my duties. I've followed the commandments. I've worked hard, worshipped you and separated waste. Yet, my dog died, my wife left me, my credit card got revoked and my kids hate me. Why, God? Why me?!?"
It's slave mentality at its fullest. Job observes his duties and believes that God owes him for that.
Funnily enough, God completely ignores Job's complains and gives a non-tangential speech about how great he, the God, is.
At the end Job gives up, accepts that God owes him nothing, embraces master morality, buys a new dog, gets a new wife, works hard once again, improves his credit score, has new kids and lives happily for ever after.